infinite determinantal measures and the ergodic decomposition … · ergodic decomposition,...

Izvestiya Mathematics 796 1111ndash1156 Izvestiya RAN Ser Mat 796 18ndash64

DOI 101070IM2015v079n06ABEH002775

Infinite determinantal measures and the ergodicdecomposition of infinite Pickrell measures

I Construction of infinite determinantal measures

A I Bufetov

Abstract This paper is the first in a series of three We give an explicitdescription of the ergodic decomposition of infinite Pickrell measures on thespace of infinite complex matrices A key role is played by the constructionof σ-finite analogues of determinantal measures on spaces of configurationsincluding the infinite Bessel process a scaling limit of the σ-finite analoguesof the Jacobi orthogonal polynomial ensembles Our main result identifiesthe infinite Bessel process with the decomposing measure of an infinitePickrell measure

Keywords determinantal processes infinite determinantal measuresergodic decomposition infinite harmonic analysis infinite unitary groupscaling limits Jacobi polynomials Harish-ChandrandashItzyksonndashZuber orbitintegral

sect 1 Introduction

11 Informal outline of main results The Pickrell family of measures is givenby the formula

micro(s)n = constns det(1 + zlowastz)minus2nminuss dz

Here n is a positive integer s a real number z an ntimesn matrix with complex entriesdz the Lebesgue measure on the space of such matrices and constns a normalizatingconstant whose precise value will be obtained below The measure micro(s)

n is finiteif s gt minus1 and infinite if s 6 minus1 It is clear from the definition that micro(s)

n is invariantunder the actions by the unitary group U(n) of multiplication on the left andon the right

This work was supported by the project AMIDEX (no ANR-11-IDEX-0001-02) of the FrenchRepublic Government Programme lsquoInvesting in the Futurersquo carried out by the French NationalAgency of Scientific Research (ANR) It was also supported by the Programme of GovernmentalSupport of Scientific Research of Young Russian Scholars Candidates and Doctors of Sciences(grant no MD-285920141) the Programme of Fundamental Research of RAS no I28P lsquoMath-ematical problems of modern control theoryrsquo (project no 0014-2015-0006 lsquoErgodic theory anddynamical systemsrsquo) the subsidy for governmental support of leading universities of the RussianFederation aimed at raising their competitiveness among world leading scientific and educationalcentres distributed to the National Research University lsquoHigher School of Economicsrsquo and theRFBR (grant no 13-01-12449-ofi m)

AMS 2010 Mathematics Subject Classification 20C32 22D40 28C10 28D15 43A05 60B1560G55

ccopy 2015 Russian Academy of Sciences (DoM) London Mathematical Society Turpion Ltd

1112 A I Bufetov

If the constants constns are chosen appropriately then the Pickrell family ofmeasures has the Kolmogorov property of consistency with respect to natural pro-jections the push-forward of the Pickrell measure micro

(s)n+1 under the natural pro-

jection sending each (n + 1) times (n + 1) matrix to its upper left n times n corner isprecisely the Pickrell measure micro(s)

n This consistency property also holds for infinitePickrell measures provided that n is sufficiently large (see Proposition 18 for anexact statement) The consistency property and Kolmogorovrsquos theorem enable oneto define the Pickrell family of measures micro(s) s isin R on the space of infinite com-plex matrices The Pickrell measures are invariant under the actions of the infiniteunitary group on the left and right and the Pickrell family of measures is the nat-ural infinite-dimensional analogue of the canonical unitarily invariant measure ona Grassmann manifold (see the paper of Pickrell [1])

What is the ergodic decomposition of Pickrell measures with respect to the actionof the Cartesian square of the infinite unitary group The ergodic unitarily invari-ant probability measures on the space of infinite complex matrices were explicitlydescribed by Pickrell [2] and another approach to their description was given byVershik and Olshanski [3] Each ergodic measure is parametrized by an infinitearray x = (x1 xn ) of numbers in the half-line satisfying x1 gt x2 gt middot middot middot gt 0and x1 + middot middot middot + xn + middot middot middot lt +infin and an additional parameter γ that we call theGaussian parameter Informally speaking the parameters xn may be thought of asthe lsquoasymptotic eigenvaluesrsquo of an infinite complex matrix and γ as the differencebetween the lsquoasymptotic tracersquo and the sum of the asymptotic eigenvalues (thisdifference is positive in particular for a Gaussian measure)

In 2000 Borodin and Olshanski [4] proved that for finite Pickrell measures theGaussian parameter vanishes almost surely and the corresponding spectral measure(the measure parametrizing the ergodic decomposition) regarded as a measure onthe space of configurations on the half-line (0+infin) coincides with the Bessel pointprocess of Tracy and Widom [5] The correlation functions of this process areexpressed as determinants of the Bessel kernel

Borodin and Olshanski [4] posed the following problem Describe the ergodicdecomposition of infinite Pickrell measures In this paper we give a solution

The first step is the result in [6] saying that almost all ergodic components of aninfinite Pickrell measure are themselves finite only the spectral measure is infiniteThen it is proved as for finite measures that the Gaussian parameter vanishesalmost surely The ergodic decomposition is thus given by a σ-finite measure B(s)

on the space of configurations on the half-line (0+infin)How to describe σ-finite measures on the space of configurations We note that

the formalism of correlation functions is completely inapplicable since they aredefined only for finite measures

This paper gives for a first time an explicit method for constructing infinitemeasures on the space of configurations Since these measures are very closelyrelated to determinantal probability measures we call them infinite determinantalmeasures

We give three descriptions of the measure B(s) The first two may be carried outin much greater generality

Ergodic decomposition of infinite Pickrell measures I 1113

(1) Inductive limit of determinantal measures By definition B(s) is sup-ported on those configurations X whose particles accumulate only to zero notto infinity Hence B(s)-almost every configuration X admits a maximal parti-cle xmax(X) By choosing an arbitrary R gt 0 and restricting B(s) to the setX xmax(X) lt R we get a finite measure which becomes determinantal afternormalization The corresponding operator is an orthogonal projection operatorwhose range can be described explicitly for all R gt 0 Thus the measure B(s)

may be obtained as an inductive limit of finite determinantal measures along anexhausting 1 family of subsets of the space of configurations

(2) A determinantal measure multiplied by a multiplicative functionalMore generally one can reduce B(s) to a finite determinantal measure by taking itsproduct with an appropriate multiplicative functional A multiplicative functionalon the space of configurations may be obtained as the product of the values ofa fixed non-negative function over all particles of a configuration

Ψg(X) =prodxisinX

g(x)

If g is suitably chosen then the measure

ΨgB(s) (1)

is finite and becomes determinantal after normalization The corresponding oper-ator is an orthogonal projection operator with explicitly computable range Ofcourse the previous description is the particular case g = χ(0R) of this It is oftenconvenient to take a strictly positive function for example gβ(x) = exp(minusβx) forβ gt 0 The range of the orthogonal projection operator inducing the measure (1)is known explicitly for a large class of functions g but an explicit formula for itskernel is known only for several concrete functions These computations will appearin a sequel to this paper

(3) A skew product As noted above B(s)-almost every configuration X con-tains a maximal particle xmax(X) Therefore it is natural for the measure B(s) to fixthe position xmax(X) of the maximal particle and consider the corresponding con-ditional measure This is a well-defined determinantal probability measure inducedby a projection operator whose range can be found explicitly (using the descrip-tion of Palm measures of determinantal point processes introduced by Shirai andTakahashi [7]) The σ-finite distribution of the maximal particle can also be foundexplicitly the ratios of the measures of intervals are obtained as ratios of suitableFredholm determinants The measure B(s) is thus represented as a skew prod-uct whose base is an explicitly known σ-finite measure on the half-line and whosefibres are certain explicitly known determinantal probability measures (see sect 110for a detailed presentation)

A key role in the construction of infinite determinantal measures is played bythe following result of [8] (see also [9]) the product of a determinantal probabilitymeasure and an integrable multiplicative functional is after normalization againa determinantal probability measure whose operator can be found explicitly In

1A family of bounded sets Bn is said to be exhausting if Bnminus1 sub Bn andS

Bn = E

1114 A I Bufetov

particular if PΠ is a determinantal point process induced by a projection operator Πwith range L then under certain additional assumptions the process ΨgPΠ is afternormalization again the determinantal point process induced by the projectionoperator onto the subspace

radicg L (a precise statement is given in Proposition B3)

Informally speaking if the function g is such that the subspaceradicg L does not lie

in L2 then the measure ΨgPΠ ceases to be finite and we obtain precisely an infinitedeterminantal measure corresponding to a subspace of locally square-integrablefunctions This is one of our main constructions (see Theorem 211)

The Bessel point process of Tracy and Widom which governs the ergodic decom-position of finite Pickrell measures is a scaling limit of Jacobi orthogonal polyno-mial emsembles To describe the ergodic decomposition of infinite Pickrell mea-sures one must consider the scaling limit of infinite analogues of Jacobi orthogonalpolynomial emsembles The resulting infinite determinantal measure is computedin this paper and is called the infinite Bessel point process (see sect 14 for a precisedefinition)

Our main result Theorem 111 identifies the ergodic decomposition measure ofan infinite Pickrell measure with the infinite Bessel point process

12 Historical remarks Pickrell measures were introduced by Pickrell [1]in 1987 In 2000 Borodin and Olshanski [4] studied a closely related two-parameterfamily of measures on the space of infinite Hermitian matrices that are invariantwith respect to the natural action of the infinite unitary group by conjugationSince the existence of such measures (as well as of the original family considered byPickrell) is proved by a computation going back to the work of Hua Loo-Keng [10]Borodin and Olshanski called their family of measures the HuandashPickrell measuresFor various generalizations of the HuandashPickrell measures see for example thepapers of Neretin [11] and Bourgade Nikehbali Rouault [12] Pickrell consideredonly those values of the parameter for which the corresponding measures are finitewhile Borodin and Olshanski [4] showed that the infinite Pickrell and HuandashPickrellmeasures are also well defined Borodin and Olshanski [4] also proved that theergodic decomposition of HuandashPickrell probability measures is given by a determi-nantal point process arising as a scaling limit of pseudo-Jacobian orthogonal poly-nomial ensembles and posed the problem of describing the ergodic decompositionof infinite HuandashPickrell measures

The aim of the present paper which is devoted to Pickrellrsquos original modelis to give an explicit description of the ergodic decomposition of infinite Pickrellmeasures on spaces of infinite complex matrices

13 Organization of the paper This paper is the first in a cycle of three givingan explicit description of the ergodic decomposition of infinite Pickrell measuresReferences to the other parts [13] [14] are organized as follows Proposition II23equation (III9) and so on

The paper is organized as follows The introduction begins with a descriptionof the main construction (infinite determinantal measures) in the concrete case ofthe infinite Bessel point process We recall the construction of Pickrell measuresand the OlshanskindashVershik approach to Pickrellrsquos classification of ergodic unitarilyequivalent measures on the space of infinite complex matrices Then we state the


main result of the paper Theorem 111 which identifies the ergodic decompositionmeasure of an infinite Pickrell measure with the infinite Bessel process (up to thechange of variable y = 4x) We conclude the introduction by giving an outline ofthe proof of Theorem 111 the ergodic decomposition measures of Pickrell measuresare obtained as scaling limits of their finite-dimensional approximations (the radialparts of finite-dimensional projections of Pickrell measures) First Lemma 114says that the rescaled radial parts multiplied by a certain density converge to thedesired ergodic decomposition measure multiplied by the same density Second itturns out that the normalized products of the push-forwards of the rescaled radialparts in the space of configurations on the half-line with an appropriately chosenmultiplicative functional on the space of configurations converge weakly in the spaceof measures on the space of configurations Combining these two facts we completethe proof of Theorem 111sect 2 is devoted to a general construction of infinite determinantal measures on

the space of configurations Conf(E) of a complete locally compact metric space Eendowed with a σ-finite Borel measure micro

We start with a space H of functions on E that are locally square integrablewith respect to micro and with an increasing family of subsets

E0 sub E1 sub middot middot middot sub En sub middot middot middot

in E such that for every n isin N the restriction χEnH is a closed subspace of L2(Emicro)

If the corresponding projection operator Πn is locally of trace class then the projec-tion operator Πn induces a determinantal measure Pn on Conf(E) by the MacchindashSoshnikov theorem Under certain additional assumptions it follows from the resultsof [8] (see Corollary B5 below) that the measures Pn satisfy the following consis-tency condition Let Conf(EEn) be the set of configurations all of whose particleslie in En Then for every n isin N we have

Pn+1|Conf(EEn)

Pn+1(Conf(EEn))= Pn (2)

The consistency property (2) implies that there is a σ-finite measure B such thatfor every n isin N we have 0 lt B(Conf(EEn)) lt +infin and

B|Conf(EEn)

B(Conf(EEn))= Pn

The measure B is called an infinite determinantal measure An alternative descrip-tion of infinite determinantal measures uses the formalism of multiplicative func-tionals It was proved in [8] (see also [9] and Proposition B3 below) that theproduct of a determinantal measure and an integrable multiplicative functional isafter normalization again a determinantal measure Taking the product of a deter-minantal measure and a convergent (but not integrable) multiplicative functionalwe obtain an infinite determinantal measure This reduction of infinite determinan-tal measures to ordinary ones by taking the product with a multiplicative functionalis essential in the proof of Theorem 111 We conclude sect 2 by proving the existenceof the infinite Bessel point process

1116 A I Bufetov

The paper has three appendices In Appendix A we collect the necessary factsabout Jacobi orthogonal polynomials including the recurrence relation betweenthe nth ChristoffelndashDarboux kernel corresponding to the parameters (α β) and the(n minus 1)th ChristoffelndashDarboux kernel corresponding to the parameters (α + 2 β)

Appendix B is devoted to determinantal point processes on spaces of configu-rations We first recall the definition of a space of configurations its Borel struc-ture and its topology Then we define determinantal point processes recall theMacchindashSoshnikov theorem and state the rule of transformation of kernels undera change of variable We also recall the definition of a multiplicative functionalon the space of configurations state the result of [8] (see also [9]) saying that theproduct of a determinantal point process and a multiplicative functional is againa determinantal point process and give an explicit representation of the resultingkernel In particular we recall the representations from [8] [9] for the kernels ofinduced processes

Appendix C is devoted to the construction of Pickrell measures following a com-putation of Hua LoondashKeng [10] and the observation of Borodin and Olshanski inthe infinite case

14 The infinite Bessel point process

141 Outline of the construction Take n isin N s isin R and endow the cube (minus1 1)n

with the measure prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)

When s gt minus1 the measure (3) is a particular case of the Jacobi orthogonal poly-nomial ensemble (a determinantal point process induced by the nth ChristoffelndashDarboux projection operator for Jacobi polynomials) The classical HeinendashMehlerasymptotics for Jacobi polynomials yields an asymptotic formula for the ChristoffelndashDarboux kernels and therefore for the corresponding determinantal point pro-cesses whose scaling limit with respect to the scaling

ui = 1minus yi

2n2 i = 1 n (4)

is the Bessel point process of Tracy and Widom [5] We recall that the Besselpoint process is governed by the operator of projection of L2((0+infin) Leb) ontothe subspace of functions whose Hankel transform is supported on [0 1]

For s 6 minus1 the measure (3) is infinite To describe its scaling limit we first recalla recurrence relation between the ChristoffelndashDarboux kernels for Jacobi polyno-mials and a corollary recurrence relations between the corresponding orthogonalpolynomial ensembles Namely the nth ChristoffelndashDarboux kernel of the Jacobiensemble with parameter s is a rank-one perturbation of the (nminus 1)th ChristoffelndashDarboux kernel of the Jacobi ensemble with parameter s+ 2

This recurrence relation motivates the following construction Consider the rangeof the ChristoffelndashDarboux projection operator This is a finite-dimensional sub-space of polynomials of degree at most n minus 1 multiplied by the weight (1 minus u)s2Consider the same subspace for s 6 minus1 Being no longer a subspace of L2 it isnevertheless a well-defined space of locally square-integrable functions In view of


the recurrence relations our subspace corresponding to a parameter s is a rank-oneperturbation of the similar subspace corresponding to the parameter s + 2 andso on until we arrive at a value of the parameter (to be denoted by s + 2ns) forwhich the subspace becomes a part of L2 Thus our initial subspace is a finite-rankperturbation of a closed subspace of L2 and the rank of this perturbation dependson s but not on n Taking the scaling limit of this representation we obtain a sub-space of locally square-integrable functions on (0+infin) which is again a finite-rankperturbation of the range of the Bessel projection operator which corresponds tothe parameter s+ 2ns

To every such subspace of locally square-integrable functions we then assigna σ-finite measure on the space of configurations the infinite Bessel point processThe infinite Bessel point process is a scaling limit of the measures (3) under thescaling (4)

142 The Jacobi orthogonal polynomial ensemble We first consider the case whens gt minus1 Let P (s)

n be the standard Jacobi orthogonal polynomials correspondingto the weight (1 minus u)s and let K(s)

n (u1 u2) be the nth ChristoffelndashDarboux ker-nel for the Jacobi orthogonal polynomial ensemble (see the formulae (34) (35) inAppendix A) For s gt minus1 we have the following well-known determinantal repre-sentation of the measure (3)

constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)

where the normalization constant constns is chosen in such a way that the left-handside is a probability measure

143 The recurrence relation for Jacobi orthogonal polynomial ensembles Wewrite Leb for the ordinary Lebesgue measure on (a subset of) the real axis Givena finite family of functions f1 fN on the line we write span(f1 fN ) forthe vector space spanned by these functions The ChristoffelndashDarboux kernel K(s)

n

is the kernel of the orthogonal projection operator of L2([minus1 1] Leb) onto thesubspace

L(sn)Jac = span

((1minus u)s2 (1minus u)s2u (1minus u)s2unminus1

)= span

((1minus u)s2 (1minus u)s2+1 (1minus u)s2+nminus1

)

By definition we have a direct-sum decomposition

L(sn)Jac = C(1minus u)s2 oplus L

(s+2nminus1)Jac

By Proposition A1 for every s gt minus1 we have the recurrence relation

K(s)n (u1 u2) =

s+ 12s+1

P(s+1)nminus1 (u1)(1minus u1)s2P

(s+1)nminus1 (u2)(1minus u2)s2 + K(s+2)

n (u1 u2)

and therefore an orthogonal direct sum decomposition

L(sn)Jac = CP (s+1)

nminus1 (u)(1minus u)s2 oplus L(s+2nminus1)Jac

1118 A I Bufetov

We now pass to the case when s 6 minus1 We define a positive integer ns by therelation

s

2+ ns isin

(minus1

212

]and consider the subspace

V (sn) = span((1minus u)s2 (1minus u)s2+1 P

(s+2nsminus1)nminusns

(u)(1minus u)s2+nsminus1) (6)

By definition we get a direct-sum decomposition

L(sn)Jac = V (sn) oplus L

(s+2nsnminusns)Jac (7)

Note thatL

(s+2nsnminusns)Jac sub L2([minus1 1] Leb)

whileV (sn) cap L2([minus1 1] Leb) = 0

144 Scaling limits We recall that the scaling limit of the ChristoffelndashDarbouxkernels K(s)

n of the Jacobi orthogonal polynomial ensemble with respect to thescaling (4) is the Bessel kernel Js of Tracy and Widom [5] (We recall a definitionof the Bessel kernel in Appendix A and give a precise statement on the scaling limitin Proposition A3)

Clearly for every β under the scaling (4) we have

limnrarrinfin

(2n2)β(1minus ui)β = yβi

and for every α gt minus1 the classical HeinendashMehler asymptotics for the Jacobipolynomials yields that

limnrarrinfin

2minusα+1

2 nminus1P (α)n (ui)(1minus ui)

αminus12 =

Jα(radicyi)radicyi

It is therefore natural to take the subspace

V (s) = span(ys2 ys2+1

Js+2nsminus1(radicy)

radicy

)as the scaling limit of the subspaces (6)

Moreover we already know that the scaling limit of the subspace (7) is the rangeL(s+2ns) of the operator Js+2ns

Thus we arrive at the subspace H(s)

H(s) = V (s) oplus L(s+2ns)

It is natural to regard H(s) as the scaling limit of the subspaces L(sn)Jac as n rarr infin

under the scaling (4)Note that the subspaces H(s) consist of functions that are locally square inte-

grable and moreover fail to be square integrable only at zero for every ε gt 0 thesubspace χ[ε+infin)H

(s) lies in L2


145 Definition of the infinite Bessel point process We now proceed to give a pre-cise description in this specific case of one of our main constructions that of theσ-finite measure B(s) the scaling limit of the infinite Jacobi ensembles (3) underthe scaling (4) Let Conf((0+infin)) be the space of configurations on (0+infin)Given any Borel subset E0 sub (0+infin) we write Conf((0+infin)E0) for the sub-space of configurations all of whose particles lie in E0 As usual given a measure Bon X and a measurable subset Y sub X with 0 lt B(Y ) lt +infin we write B|Y for therestriction of B to Y

It will be proved in what follows that for every ε gt 0 the subspace χ(ε+infin)H(s) is

a closed subspace of L2((0+infin) Leb) and that the orthogonal projection operatorΠ(εs) onto χ(ε+infin)H

(s) is locally of trace class By the MacchindashSoshnikov theoremthe operator Π(εs) induces a determinantal measure PeΠ(εs) on Conf((0+infin))

Proposition 11 Let s 6 minus1 Then there is a σ-finite measure B(s) onConf((0+infin)) such that the following conditions hold

(1) The particles of B-almost all configurations do not accumulate at zero(2) For every ε gt 0 we have

0 lt B(Conf((0+infin) (ε+infin))

)lt +infin

B|Conf((0+infin)(ε+infin))

B(Conf((0+infin) (ε+infin)))= PeΠ(εs)

These conditions uniquely determine the measure B(s) up to multiplication bya constant

Remark When s 6= minus1minus3 we can also write

H(s) = span(ys2 ys2+nsminus1)oplus L(s+2ns)

and use the previous construction without any further changes Note that whens = minus1 the function y12 is not square integrable at infinity whence the need forthe definition above When s gt minus1 we put B(s) = PJs

Proposition 12 If s1 6= s2 then the measures B(s1) and B(s2) are mutuallysingular

The proof of Proposition 12 will be deduced from Proposition 14 which willin turn be obtained from our main result Theorem 111

15 The modified Bessel point process In what follows we use the Besselpoint process subject to the change of variable y = 4x To describe it we considerthe half-line (0+infin) with the standard Lebesgue measure Leb Take s gt minus1 andintroduce a kernel J (s) by the formula

J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt

The change of variable y = 4x reduces the kernel J (s) to the kernel Js ofthe Bessel point process of Tracy and Widom as considered above (we recall thata change of variables u1 = ρ(v1) u2 = ρ(v2) transforms a kernel K(u1 u2) toa kernel of the form K(ρ(v1) ρ(v2))

radicρprime(v1)ρprime(v2)) Thus the kernel J (s) induces

a locally trace-class orthogonal projection operator on L2((0+infin) Leb) Slightlyabusing our notation we denote this operator again by J (s) Let L(s) be the rangeof J (s) By the MacchindashSoshnikov theorem the operator J (s) induces a determi-nantal measure PJ(s) on the space of configurations Conf((0+infin))

16 The modified infinite Bessel point process The involutive homeomor-phism y = 4x of the half-line (0+infin) induces the corresponding homeomorphism(change of variable) of the space Conf((0+infin)) Let B(s) be the image of B(s)

under the change of variables We shall see below that B(s) is precisely the ergodicdecomposition measure for the infinite Pickrell measures

A more explicit description of B(s) can be given as followsBy definition we put

L(s) =ϕ(4x)x

ϕ isin L(s)

(The behaviour of determinantal measures under a change of variables is describedin sectB5)

We similarly write V (s)H(s) sub L2loc((0+infin) Leb) for the images of the sub-spaces V (s) H(s) under the change of variable y = 4x

V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)

By definition we have

V (s) = span(xminuss2minus1

Js+2nsminus1(2radicx)radic

x

) H(s) = V (s) oplus L(s+2ns)

We shall see that for allRgt0 χ(0R)H(s) is a closed subspace of L2((0+infin) Leb)

Let Π(sR) be the corresponding orthogonal projection By definition Π(sR) islocally of trace class and by the MacchindashSoshnikov theorem Π(sR) induces a deter-minantal measure PΠ(sR) on Conf((0+infin))

The measure B(s) is characterized by the following conditions(1) The set of particles of B(s)-almost every configuration is bounded(2) For every R gt 0 we have

0 lt B(Conf((0+infin) (0 R))

)lt +infin

B|Conf((0+infin)(0R))

B(Conf((0+infin) (0 R)))= PΠ(sR)

These conditions uniquely determine B(s) up to a constant



H(s) = span(xminuss2minus1 xminuss2minusns+1)oplus L(s+2ns)

Let I1loc((0+infin) Leb) be the space of locally trace-class operators on the spaceL2((0+infin) Leb) (see sectB3 for a detailed definition) The following propositiondescribes the asymptotic behaviour of the operators Π(sR) as Rrarrinfin

Proposition 13 Let s 6 minus1 Then the following assertions hold(1) As Rrarrinfin we have Π(sR) rarr J (s+2ns) in I1loc((0+infin) Leb)(2) The corresponding measures also converge as Rrarrinfin

PΠ(sR) rarr PJ(s+2ns)

weakly in the space of probability measures on Conf((0+infin))

As above for s gt minus1 we put B(s) = PJ(s) Proposition 12 is equivalent to thefollowing assertion


We will obtain Proposition 14 in the last section of the paper as a corollary ofour main result Theorem 111

We now represent the measure B(s) as the product of a determinantal probabil-ity measure and a multiplicative functional Here we restrict ourselves to a specificexample of such representation but we shall see in what follows that the construc-tion holds in much greater generality We introduce a function S on the space ofconfigurations Conf((0+infin)) putting

S(X) =sumxisinX

x

Of course the function S may take the value infin but the following propositionshows that the set of such configurations has B(s)-measure zero

Proposition 15 For every s isin R we have S(X) lt +infin almost surely with respectto the measure B(s) and for any β gt 0

exp(minusβS(X)

)isin L1

(Conf((0+infin)) B(s)

)

Furthermore we shall now see that the measure

exp(minusβS(X))B(s)intConf((0+infin))

exp(minusβS(X)) dB(s)

is determinantal

Proposition 16 For all s isin R and β gt 0 the subspace

exp(minusβx

2

)H(s) (8)

is a closed subspace of L2

((0+infin) Leb

)and the operator of orthogonal projection

onto (8) is locally of trace class

1122 A I Bufetov

Let Π(sβ) be the operator of orthogonal projection onto the subspace (8)By Proposition 16 and the MacchindashSoshnikov theorem the operator Π(sβ)

induces a determinantal probability measure on the space Conf((0+infin))

Proposition 17 For all s isin R and β gt 0 we have


exp(minusβS(X)) dB(s)= PΠ(sβ) (9)

17 Unitarily invariant measures on spaces of infinite matrices

171 Pickrell measures Let Mat(nC) be the space of complex n times n matrices

Mat(nC) =z = (zij) i = 1 n j = 1 n

Let Leb = dz be the Lebesgue measure on Mat(nC) For n1 lt n let

πnn1

Mat(nC) rarr Mat(n1C)

be the natural projection sending each matrix z = (zij) i j = 1 n to its upperleft corner that is the matrix πn

n1(z) = (zij) i j = 1 n1

Following Pickrell [2] we take s isin R and introduce a measure micro(s)n on Mat(nC)

by the formulamicro(s)

n = det(1 + zlowastz)minus2nminussdz

The measure micro(s)n is finite if and only if s gt minus1

The measures micro(s)n have the following property of consistency with respect to the

projections πnn1

Proposition 18 Let s isin R and n isin N be such that n + s gt 0 Then for everymatrix z isin Mat(nC) we haveint

(πn+1n )minus1(z)

det(1+zlowastz)minus2nminus2minuss dz =π2n+1(Γ(n+ 1 + s))2

Γ(2n+ 2 + s)Γ(2n+ 1 + s)det(1+zlowastz)minus2nminuss

We now consider the space Mat(NC) of infinite complex matrices whose rowsand columns are indexed by positive integers

Mat(NC) =z = (zij) i j isin N zij isin C

Let πinfinn Mat(NC) rarr Mat(nC) be the natural projection sending each infinitematrix z isin Mat(NC) to its upper left n times n corner that is the matrix (zij)i j = 1 n

For s gt minus1 it follows from Proposition 18 and Kolmogorovrsquos existence theo-rem [15] that there is a unique probability measure micro(s) on Mat(NC) such that forevery n isin N we have

(πinfinn )lowastmicro(s) = πminusn2nprod

l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n


If s 6 minus1 then Proposition 18 along with Kolmogorovrsquos existence theorem [15]enables us to conclude that for every λ gt 0 there is a unique infinite measure micro(sλ)

on Mat(NC) with the following properties(1) For every n isin N satisfying n+ s gt 0 and every compact set Y sub Mat(nC)

we have micro(sλ)(Y ) lt +infin Hence the push-forwards (πinfinn )lowastmicro(sλ) are well defined(2) For every n isin N satisfying n+ s gt 0 we have

(πinfinn )lowastmicro(sλ) = λ

( nprodl=n0

πminus2n Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

)micro(s)

The measures micro(sλ) are called infinite Pickrell measures Slightly abusing ournotation we shall omit the superscript λ and write micro(s) for a measure defined upto a multiplicative constant A detailed definition of infinite Pickrell measures isgiven by Borodin and Olshanski [4] p 116

Proposition 19 For all s1 s2 isin R with s1 6= s2 the Pickrell measures micro(s1) andmicro(s2) are mutually singular

Proposition 19 may be obtained from Kakutanirsquos theorem in the spirit of [4](see also [11])

Let U(infin) be the infinite unitary group An infinite matrix u = (uij)ijisinN belongsto U(infin) if there is a positive integer n0 such that the matrix (uij) i j isin [1 n0]is unitary uii = 1 for i gt n0 and uij = 0 whenever i 6= j max(i j) gt n0 Thegroup U(infin)timesU(infin) acts on Mat(NC) by multiplication on both sides T(u1u2)z =u1zu

minus12 The Pickrell measures micro(s) are by definition invariant under this action

The role of Pickrell (and related) measures in the representation theory of U(infin) isreflected in [3] [16] [17]

It follows from Theorem 1 and Corollary 1 in [18] that the measures micro(s) admitan ergodic decomposition Theorem 1 in [6] says that for every s isin R almost allergodic components of micro(s) are finite We now state this result in more detailRecall that a (U(infin)timesU(infin))-invariant measure on Mat(NC) is said to be ergodicif every (U(infin)timesU(infin))-invariant Borel subset of Mat(NC) either has measure zeroor has a complement of measure zero Equivalently ergodic probability measuresare extreme points of the convex set of all (U(infin) times U(infin))-invariant probabilitymeasures on Mat(NC) We denote the set of all ergodic (U(infin)timesU(infin))-invariantprobability measures on Mat(NC) by Merg(Mat(NC)) The set Merg(Mat(NC))is a Borel subset of the set of all probability measures on Mat(NC) (see for exam-ple [18]) Theorem 1 in [6] says that for every s isin R there is a unique σ-finite Borelmeasure micro(s) on Merg(Mat(NC)) such that

micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)

Our main result is an explicit description of the measure micro(s) and its identifica-tion after a change of variable with the infinite Bessel point process consideredabove

1124 A I Bufetov

18 Classification of ergodic measures We recall a classification of ergodic(U(infin) times U(infin))-invariant probability measures on Mat(NC) This classificationwas obtained by Pickrell [2] [19] Vershik [20] and Vershik and Olshanski [3] sug-gested another approach to it in the case of unitarily invariant measures on the spaceof infinite Hermitian matrices and Rabaoui [21] [22] adapted the VershikndashOlshanskiapproach to the original problem of Pickrell We also follow the approach of Vershikand Olshanski

We take a matrix z isin Mat(NC) put z(n) = πinfinn z and let

λ(n)1 gt middot middot middot gt λ(n)

n gt 0

be the eigenvalues of the matrix

(z(n))lowastz(n)

counted with multiplicities and arranged in non-increasing order To stress thedependence on z we write λ(n)

i = λ(n)i (z)

Theorem (1) Let η be an ergodic (U(infin) times U(infin))-invariant Borel probabilitymeasure on Mat(NC) Then there are non-negative real numbers

γ gt 0 x1 gt x2 gt middot middot middot gt xn gt middot middot middot gt 0

satisfying the condition γ gtsuminfin

i=1 xi and such that for η-almost all matrices z isinMat(NC) and every i isin N we have

xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)

(2) Conversely suppose we are given any non-negative real numbers γ gt 0x1 gt x2 gt middot middot middot gt xn gt middot middot middot gt 0 such that γ gt

suminfini=1 xi Then there is a unique

ergodic (U(infin) times U(infin))-invariant Borel probability measure η on Mat(NC) suchthat the relations (10) hold for η-almost all matrices z isin Mat(NC)

We define the Pickrell set ΩP sub R+ times RN+ by the formula

ΩP =ω = (γ x) x = (xn) n isin N xn gt xn+1 gt 0 γ gt

infinsumi=1

xi

By definition ΩP is a closed subset of the product R+ times RN+ endowed with the

Tychonoff topology For ω isin ΩP let ηω be the corresponding ergodic probabilitymeasure

The Fourier transform of ηω may be described explicitly as follows For everyλ isin R we haveint

Mat(NC)

exp(iλRe z11) dηω(z) =exp

(minus4

(γ minus

suminfink=1 xk

)λ2

)prodinfink=1(1 + 4xkλ2)

(11)

We denote the right-hand side of (11) by Fω(λ) Then for all λ1 λm isin R wehaveint

Mat(NC)

exp(i(λ1 Re z11 + middot middot middot+ λm Re zmm)

)dηω(z) = Fω(λ1) middot middot Fω(λm)


Thus the Fourier transform is completely determined and therefore the measureηω is completely described

An explicit construction of the measures ηω is as follows Letting all entriesof z be independent identically distributed complex Gaussian random variableswith expectation 0 and variance γ we get a Gaussian measure with parameter γClearly this measure is unitarily invariant and by Kolmogorovrsquos zero-one lawergodic It corresponds to the parameter ω = (γ 0 0 ) all x-coordinatesare equal to zero (indeed the eigenvalues of a Gaussian matrix grow at the rateofradicn rather than n) We now consider two infinite sequences (v1 vn ) and

(w1 wn ) of independent identically distributed complex Gaussian randomvariables with variance

radicx and put zij = viwj This yields a measure which

is clearly unitarily invariant and by Kolmogorovrsquos zero-one law ergodic Thismeasure corresponds to a parameter ω isin ΩP such that γ(ω) = x x1(ω) = xand all other parameters are equal to zero Following Vershik and Olshanski [3]we call such measures Wishart measures with parameter x In the general case weput γ = γ minus

suminfink=1 xk The measure ηω is then an infinite convolution of the

Wishart measures with parameters x1 xn and a Gaussian measure withparameter γ This convolution is well defined since the series x1 + middot middot middot + xn + middot middot middotconverges

The quantity γ = γ minussuminfin

k=1 xk will therefore be called the Gaussian parameterof the measure ηω We shall prove that the Gaussian parameter vanishes for almostall ergodic components of Pickrell measures

By Proposition 3 in [18] the set of ergodic (U(infin) times U(infin))-invariant mea-sures is a Borel subset in the space of all Borel measures on Mat(NC) endowedwith the natural Borel structure (see for example [23]) Furthermore writingηω for the ergodic Borel probability measure corresponding to a point ω isin ΩP ω = (γ x) we see that the correspondence ω rarr ηω is a Borel isomorphism betweenthe Pickrell set ΩP and the set of ergodic (U(infin) times U(infin))-invariant probabilitymeasures on Mat(NC)

The ergodic decomposition theorem (Theorem 1 and Corollary 1 in [18]) saysthat each Pickrell measure micro(s) s isin R induces a unique decomposition measure micro(s)

on ΩP such that

micro(s) =int

ΩP

ηω dmicro(s)(ω) (12)

The integral is understood in the ordinary weak sense (see [18])When s gt minus1 the measure micro(s) is a probability measure on ΩP When s 6 minus1

it is infiniteWe put

Ω0P =

(γ xn) isin ΩP xn gt 0 for all n γ =

infinsumn=1

xn

Of course the subset Ω0P is not closed in ΩP We introduce the map

conf ΩP rarr Conf((0+infin))

1126 A I Bufetov

sending each point ω isin ΩP ω = (γ xn) to the configuration

conf(ω) = (x1 xn ) isin Conf((0+infin))

The map ω rarr conf(ω) is bijective when restricted to Ω0P

Remark The map conf is defined by counting multiplicities of the lsquoasymptoticeigenvaluesrsquo xn Moreover if xn0 =0 for some n0 then xn0 and all subsequent termsare discarded and the resulting configuration is finite Nevertheless we shall see thatmicro(s)-almost all configurations are infinite and micro(s)-almost surely all multiplicitiesare equal to one We shall also prove that ΩP Ω0

P has micro(s)-measure zero for all s

19 Statement of the main result We first state an analogue of the BorodinndashOlshanski ergodic decomposition theorem [4] for finite Pickrell measures

Proposition 110 Suppose that s gt minus1 Then micro(s)(Ω0P ) = 1 and the map ω rarr

conf(ω) which is bijective micro(s)-almost everywhere identifies the measure micro(s) withthe determinantal measure PJ(s)

Our main result is the following explicit description of the ergodic decompositionfor infinite Pickrell measures

Theorem 111 Suppose that s isin R and let micro(s) be the decomposition measure(defined in (12)) of the Pickrell measure micro(s) Then the following assertions hold

(1) micro(s)(ΩP Ω0P ) = 0

(2) The map ω rarr conf(ω) which is bijective micro(s)-almost everywhere identifiesthe measure micro(s) with the infinite determinantal measure B(s)

110 A skew-product representation of the measure B(s) Note that B(s)-almost all configurations X may accumulate only at zero and therefore admita maximal particle which we denote by xmax(X) We are interested in the distri-bution of values of xmax(X) with respect to B(s) By definition for every R gt 0B(s) takes finite values on the sets X xmax(X) lt R Furthermore again bydefinition the following relation holds for R gt 0 and R1 R2 6 R

B(s)(X xmax(X) lt R1)B(s)(X xmax(X) lt R2)

=det(1minus χ(R1+infin)Π(sR)χ(R1+infin))det(1minus χ(R2+infin)Π(sR)χ(R2+infin))

The push-forward of the measure B(s) is a well-defined σ-finite Borel measureon (0+infin) We denote it by ξmaxB(s) Of course the measure ξmaxB(s) is definedup to multiplication by a positive constant

Question What is the asymptotic behaviour of ξmaxB(s)(0 R) as R rarr infin andas Rrarr 0

We again denote the kernel of the operator Π(sR) by Π(sR) Consider thefunction ϕR(x) = Π(sR)(xR) By definition

ϕR(x) isin χ(0R)H(s)

Let H(sR) be the orthogonal complement of the one-dimensional subspace spannedby ϕR(x) in χ(0R)H

(s) In other words H(sR) is the subspace of all functions


in χ(0R)H(s) that vanish at R Let Π(sR) be the operator of orthogonal projection

onto H(sR)

Proposition 112 We have

B(s) =int infin

0

PΠ(sR) dξmaxB(s)(R)

Proof This follows immediately from the definition of B(s) and the characterizationof Palm measures for determinantal point processes (see the paper [24] by Shiraiand Takahashi)

111 The general scheme of ergodic decomposition

1111 Approximation Let F be the family of σ-finite (U(infin) times U(infin))-invariantmeasures micro on Mat(NC) for which there is a positive integer n0 (depending on micro)such that for all R gt 0 we have

micro(z max

16ij6n0|zij | lt R

)lt +infin

By definition all Pickrell measures belong to the class FWe recall a result of [6] every ergodic measure of class F is finite and therefore

the ergodic components of any measure in F are finite almost surely (The existenceof an ergodic decomposition for every measure micro isin F follows from the ergodicdecomposition theorem for actions of inductively compact groups that is inductivelimits of compact groups established in [18]) The classification of finite ergodicmeasures now yields that for every measure micro isin F there is a unique σ-finite Borelmeasure micro on the Pickrell set ΩP such that

micro =int

ΩP

ηω dmicro(ω) (13)

Our next aim is to construct following Borodin and Olshanski [4] a sequence offinite-dimensional approximations of micro

With every matrix z isin Mat(NC) and number n isin N we associate the sequence

(λ(n)1 λ

(n)2 λ(n)

n )

of all eigenvalues of the matrix (z(n))lowastz(n) arranged in non-increasing order Here

z(n) = (zij)ij=1n

For n isin N we define the map

r(n) Mat(NC) rarr ΩP

by the formula

r(n)(z) =(

1n2

tr(z(n))lowastz(n)λ

(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov

It is clear from the definition that for all n isin N and z isin Mat(NC) we have

r(n)(z) isin Ω0P

For every measure micro isin F and all sufficiently large n isin N the push-forwards(r(n))lowastmicro are well defined since the unitary group is compact We now prove that forany micro isin F the measures (r(n))lowastmicro approximate the ergodic decomposition measure micro

We start with the direct description of the map sending each measure micro isin F toits ergodic decomposition measure micro

Following Borodin and Olshanski [4] we consider the set Matreg(NC) of allregular matrices z that is matrices with the following properties

(1) For every k the limit limnrarrinfin1

n2λ(k)n = xk(z) exists

(2) The limit limnrarrinfin1

n2 tr(z(n))lowastz(n) = γ(z) existsSince the set of regular matrices has full measure with respect to any finite

ergodic (U(infin) times U(infin))-invariant measure the existence of the ergodic decompo-sition (13) implies that

micro(Mat(NC) Matreg(NC)

)= 0

We define a mapr(infin) Matreg(NC) rarr ΩP

by the formula

r(infin)(z) =(γ(z) x1(z) x2(z) xk(z)

)

The ergodic decomposition theorem [18] and the classification of ergodic unitarilyinvariant measures (in the form of Vershik and Olshanski) yield the importantequality

(r(infin))lowastmicro = micro (14)

Remark This equality has a simple analogue in the context of De Finettirsquos theoremTo obtain the ergodic decomposition of an exchangeable measure on the space ofbinary sequences it suffices to consider the push-forward of the initial measureunder the almost surely defined map sending each sequence to the frequency ofzeros in it

Given a complete separable metric space Z we write Mfin(Z) for the space of allfinite Borel measures on Z endowed with the weak topology We recall (see [23])that Mfin(Z) is itself a complete separable metric space the weak topology isinduced for example by the LevyndashProkhorov metric

We now prove that the measures (r(n))lowastmicro approximate the measure (r(infin))lowastmicro = microas nrarrinfin For finite measures micro the following result was obtained by Borodin andOlshanski [4]

Proposition 113 Let micro be a finite unitarily invariant measure on Mat(NC)Then as nrarrinfin we have

(r(n))lowastmicrorarr (r(infin))lowastmicro

weakly in Mfin(ΩP )


Proof Let f ΩP rarr R be continuous and bounded By definition for every infinitematrix z isin Matreg(NC) we have r(n)(z) rarr r(infin)(z) as nrarrinfin and therefore

limnrarrinfin

f(r(n)(z)) = f(r(infin)(z))

Hence by the dominated convergence theorem we have

limnrarrinfin

intMat(NC)

f(r(n)(z)) dmicro(z) =int

Mat(NC)

f(r(infin)(z)) dmicro(z)

Changing variables we arrive at the convergence

limnrarrinfin

intΩP

f(ω) d(r(n))lowastmicro =int

ΩP

f(ω) d(r(infin))lowastmicro

This establishes the desired weak convergence

For σ-finite measures micro isin F the result of Borodin and Olshanski can be modifiedas follows

Lemma 114 Let micro isin F Then there is a positive bounded continuous function fon the Pickrell set ΩP such that the following conditions hold

(1) f isinL1(ΩP (r(infin))lowastmicro) and f isinL1(ΩP (r(n))lowastmicro) for all sufficiently large nisinN(2) As nrarrinfin we have

f(r(n))lowastmicrorarr f(r(infin))lowastmicro


A proof of Lemma 114 will be given in the third part of this paper

Remark The argument above shows that the explicit characterization of the ergodicdecomposition of Pickrell measures in Theorem 111 relies on the abstract result(Theorem 1 in [18]) that a priori guarantees the existence of the ergodic decom-position Theorem 111 does not give an alternative proof of the existence of thisergodic decomposition

1112 Convergence of probability measures on the Pickrell set We recall the def-inition of a natural lsquoforgetfulrsquo map


This map sends each point ω = (γ x) x = (x1 xn ) to the configurationconf(ω) = (x1 xn )

For ω isin ΩP ω = (γ x) x = (x1 xn ) xn = xn(ω) we put

S(ω) =infinsum

n=1

xn(ω)

In other words we put S(ω) = S(conf(ω)) and slightly abusing the notationdenote both maps by the same letter Take β gt 0 and for every n isin N consider themeasures

exp(minusβS(ω))r(n)(micro(s))

1130 A I Bufetov


exp(minusβS(ω)) isin L1(ΩP r(n)(micro(s)))

Introduce probability measures

ν(snβ) =exp(minusβS(ω))r(n)(micro(s))int

ΩPexp(minusβS(ω)) dr(n)(micro(s))

We now reconsider the probability measure PΠ(sβ) on the space Conf((0+infin))(see (9)) and define a measure ν(sβ) on ΩP by the following requirements

(1) ν(sβ)(ΩP Ω0P ) = 0

(2) conflowast ν(sβ) = PΠ(sβ) The following proposition plays a key role in the proof of Theorem 111

Proposition 116 For all β gt 0 and s isin R we have

ν(snβ) rarr ν(sβ)

weakly on Mfin(ΩP ) as nrarrinfin

Proposition 116 will be proved in sectsect III3 III4 Combining Proposition 116with Lemma 114 we shall complete the proof of our main result Theorem 111

To prove the weak convergence of the measures ν(snβ) we first study scalinglimits of the radial parts of finite-dimensional projections of infinite Pickrell mea-sures

112 The radial part of a Pickrell measure Following Pickrell we associatewith every matrix z isin Mat(nC) the tuple (λ1(z) λn(z)) of eigenvalues of zlowastzarranged in non-decreasing order We introduce the map

radn Mat(nC) rarr Rn+

by the formularadn z rarr

(λ1(z) λn(z)

) (15)

The map (15) extends naturally to Mat(NC) and we denote this extension againby radn Thus the map radn sends each infinite matrix to the tuple of squaredeigenvalues of its upper left corner of size ntimes n

We now define the radial part of the Pickrell measure micro(s)n as the push-forward

of micro(s)n under the map radn Since the finite-dimensional unitary groups are compact

and by definition micro(s)n is finite on compact sets for every s and all sufficiently

large n the push-forward is well defined for all sufficiently large n even when themeasure micro(s) is infinite

Slightly abusing the notation we write dz for the Lebesgue measure on Mat(nC)and dλ for the Lebesgue measure on Rn

+For the push-forward of the Lebesgue measure Leb(n) = dz under the map radn

we now have(radn)lowast(dz) = const(n)

prodiltj

(λi minus λj)2 dλ


(see for example [25] [26]) where const(n) is a positive constant depending onlyon n

Then the radial part of micro(s)n takes the form

(radn)lowastmicro(s)n = const(n s)

prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ

where const(n s) is a positive constant depending only on n and sFollowing Pickrell we introduce new variables u1 un by the formula

ui =xi minus 1xi + 1

(16)

Proposition 117 In the coordinates (16) the radial part (radn)lowastmicro(s)n of the mea-

sure micro(s)n is given on the cube [minus1 1]n by the formula


prodiltj

(ui minus uj)2nprod

i=1

(1minus ui)s dui (17)

(the constant const(n s) may change from one formula to another)

If s gt minus1 then the constant const(n s) may be chosen in such a way thatthe right-hand side is a probability measure If s 6 minus1 then there is no canonicalnormalization and the left-hand side is defined up to an arbitrary positive constant

When sgtminus1 Proposition 117 yields a determinantal representation for the radialpart of the Pickrell measure Namely the radial part is identified with the Jacobiorthogonal polynomial ensemble in the coordinates (16) Passing to a scaling limitwe obtain the Bessel point process up to the change of variable y = 4x

We shall similarly prove that when s 6 minus1 the scaling limit of the measures (17)is equal to the modified infinite Bessel point process introduced above Furthermoreif we multiply the measures (17) by the density exp(minusβS(X)n2) then the resultingmeasures are finite and determinantal and their weak limit after an appropriatechoice of scaling is equal to the determinantal measure PΠ(sβ) as given in (9) Thisweak convergence is a key step in the proof of Proposition 116

Thus the study of the case s 6 minus1 requires a new object infinite determinantalmeasures on the space of configurations In the next section we proceed to the gen-eral construction and description of properties of infinite determinantal measures

Grigori Olshanski posed the problem to me and I am greatly indebted tohim I an deeply grateful to Alexei M Borodin Yanqi Qiu Klaus Schmidt andMaria V Shcherbina for useful discussions

sect 2 Construction and properties of infinite determinantal measures

21 Preliminary remarks on σ-finite measures Let Y be a Borel space Weconsider a representation

Y =infin⋃

n=1

Yn

1132 A I Bufetov

of Y as a countable union of an increasing sequence of subsets Yn Yn sub Yn+1As above given a measure micro on Y and a subset Y prime sub Y we write micro|Y prime for therestriction of micro to Y prime Suppose that for every n we are given a probability measurePn on Yn The following proposition is obvious

Proposition 21 A σ-finite measure B on Y with

B|Yn

B(Yn)= Pn (18)

exists if and only if for all N and n with N gt n we have

PN |Yn

PN (Yn)= Pn

The condition (18) uniquely determines the measure B up to multiplication by a con-stant

Corollary 22 If B1 B2 are σ-finite measures on Y such that for all n isin N wehave

0 lt B1(Yn) lt +infin 0 lt B2(Yn) lt +infin

andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)

then there is a positive constant C gt 0 such that B1 = CB2

22 The unique extension property

221 Extension from a subset Let E be a standard Borel space micro a σ-finitemeasure on E L a closed subspace of L2(Emicro) Π the operator of orthogonalprojection onto L and E0 sub E a Borel subset We say that the subspace L has theproperty of unique extension from E0 if every function ϕ isin L satisfying χE0ϕ = 0must be identically equal to zero and the subspace χE0L is closed In general thecondition that every function ϕ isin L satisfying χE0ϕ = 0 is always the zero functiondoes not imply that the restricted subspace χE0L is closed Nevertheless we havethe following obvious corollary of the open mapping theorem

Proposition 23 Let L be a closed subspace such that every function ϕ isin L withχE0ϕ = 0 is the zero function In this case the subspace χE0L is closed if and onlyif there is an ε gt 0 such that for all ϕ isin L we have

χEE0ϕ 6 (1minus ε)ϕ (19)

If this condition holds then the natural restriction map ϕ rarr χE0ϕ is an isomor-phism of Hilbert spaces If the operator χEE0Π is compact then the condition (19)holds

Remark In particular the condition (19) holds a fortiori if the operator χEE0Πis HilbertndashSchmidt or equivalently if the operator χEE0ΠχEE0 belongs to thetrace class


We immediately obtain some corollaries of Proposition 23

Corollary 24 Let g be a bounded non-negative Borel function on E such that

infxisinE0

g(x) gt 0 (20)

If the condition (19) holds then the subspaceradicg L is closed in L2(Emicro)

Remark The apparently superfluous square root is put here to keep the notationconsistent with other results in this paper

Corollary 25 Under the hypotheses of Proposition 23 if (19) holds and theBorel function g E rarr [0 1] satisfies (20) then the operator Πg of orthogonalprojection onto

radicg L is given by the formula

Πg =radicgΠ(1 + (g minus 1)Π)minus1radicg =

radicgΠ(1 + (g minus 1)Π)minus1Π

radicg (21)

In particular the operator ΠE0 of orthogonal projection onto the subspace χE0Ltakes the form

ΠE0 = χE0Π(1minus χEE0Π)minus1χE0 = χE0Π(1minus χEE0Π)minus1ΠχE0 (22)

Corollary 26 Under the hypotheses of Proposition 23 suppose that the condition(19) holds Then for every subset Y sub E0 whenever the operator χY ΠE0χY belongsto the trace class so does the operator χY ΠχY and we have

trχY ΠE0χY gt trχY ΠχY

Indeed it is clear from (22) that if the operator χY ΠE0 is HilbertndashSchmidt thenso is χY Π The inequality between traces is also immediate from (22)

222 Examples the Bessel kernel and the modified Bessel kernel

Proposition 27 (1) For every ε gt 0 the operator Js has the property of uniqueextension from (ε+infin)

(2) For every R gt 0 the operator J (s) has the property of unique extension from(0 R)

Proof Part (1) follows directly from the uncertainty principle for the Hankel trans-form a function and its Hankel transform cannot both be supported on a set offinite measure [27] [28] (note that the uncertainty principle in [27] is stated only fors gt minus12 but the more general uncertainty principle in [28] is directly applicablewhen s isin [minus1 12] see also [29]) and the following bound which clearly holds bythe definitions for every R gt 0int R

0

Js(y y) dy lt +infin

The second part follows from the first by the change of variable y = 4x

1134 A I Bufetov

23 Inductively determinantal measures Let E be a locally compact metricspace and Conf(E) the space of configurations on E endowed with the natural Borelstructure (see for example [30] [31] and sectB1 below)

Given a Borel subset Eprime sub E we write Conf(EEprime) for the subspace of thoseconfigurations all of whose particles lie in Eprime Given a measure B on X and a mea-surable subset Y sub X with 0 lt B(Y ) lt +infin we write B|Y for the restriction of Bto Y

Let micro be a σ-finite Borel measure on ETake a Borel subset E0 sub E and assume that for every bounded Borel subset

B sub E E0 we are given a closed subspace LE0cupB sub L2(Emicro) such that thecorresponding projection operator ΠE0cupB belongs to I1loc(Emicro) We also makethe following assumption

Assumption 1 (1) χBΠE0cupB lt 1 χBΠE0cupBχB isin I1(Emicro)(2) For any subsets B(1) sub B(2) sub E E0 we have

χE0cupB(1)LE0cupB(2)= LE0cupB(1)

Proposition 28 Under these assumptions there is a σ-finite measure B onConf(E) with the following properties

(1) For B-almost all configurations only finitely many particles lie in E E0(2) For every bounded Borel subset BsubEE0 we have 0 lt B(Conf(E E0cupB)) lt

+infin andB|Conf(EE0cupB)

B(Conf(EE0 cupB))= PΠE0cupB

We call such a measure B an inductively determinantal measureProposition 28 follows immediately from Proposition 21 combined with Propo-

sition B3 and Corollary B5 Note that the conditions (1) and (2) determine themeasure uniquely up to multiplication by a constant

We now give a sufficient condition for an inductively determinantal measure tobe an actual finite determinantal measure

Proposition 29 Consider a family of projections ΠE0cupB satisfying Assumption 1and let B be the corresponding inductively determinantal measure If there areR gt 0 and ε gt 0 such that for all bounded Borel subsets B sub E E0 we have

(1) χBΠE0cupB lt 1minus ε(2) trχBΠE0cupBχB lt R

then there is an operator Π isin I1loc(Emicro) of projection onto a closed subspaceL sub L2(Emicro) with the following properties

(1) LE0cupB = χE0cupBL for all B(2) χEE0ΠχEE0 isin I1(Emicro)(3) the measures B and PΠ coincide up to multiplication by a constant

Proof By our assumptions for every bounded Borel subset B sub E E0 we havea closed subspace LE0cupB (the range of the operator ΠE0cupB) which has the propertyof unique extension from E0 The uniform bounds for the norms of the operatorsχBΠE0cupB imply the existence of a closed subspace L such that LE0cupB = χE0cupBL


Since by assumption the projection operator ΠE0cupB belongs to I1loc(Emicro) weobtain for every bounded subset Y sub E that

χY ΠE0cupY χY isin I1(Emicro)

Using Corollary 26 for the subset E0 cup Y we now obtain that

χY ΠχY isin I1(Emicro)

Thus the operator Π of orthogonal projection onto L is locally of trace class andtherefore induces a unique determinantal probability measure PΠ on Conf(E)Using Corollary 26 again we obtain that

trχEE0ΠχEE0 6 R

We now give sufficient conditions for the measure B to be infinite

Proposition 210 If at least one of the following assumptions holds then themeasure B is infinite

(1) For every ε gt 0 there is a bounded Borel subset B sub E E0 such that

χBΠE0cupB gt 1minus ε

(2) For every R gt 0 there is a bounded Borel subset B sub E E0 such that

trχBΠE0cupBχB gt R

Proof We recall that

B(Conf(EE0))B(Conf(EE0 cupB))

= PΠE0cupB (Conf(EE0)) = det(1minus χBΠE0cupBχB)

If the first assumption holds then it follows immediately that the top eigenvalueof the self-adjoint trace-class operator χBΠE0cupBχB is greater than 1 minus ε whence

det(1minus χBΠE0cupBχB) 6 ε

If the second assumption holds then

det(1minus χBΠE0cupBχB) 6 exp(minus trχBΠE0cupBχB) 6 exp(minusR)

In both cases the ratioB(Conf(EE0))

B(Conf(E E0 cupB))

can be made arbitrarily small by an appropriate choice of the subset B It followsthat the measure B is infinite

1136 A I Bufetov

24 General construction of infinite determinantal measures By theMacchindashSoshnikov theorem under some additional assumptions one can constructa determinantal measure from an operator of orthogonal projection or equivalentlyfrom a closed subspace of L2(Emicro) In a similar way an infinite determinantal mea-sure may be assigned to every subspace H of locally square-integrable functions

We recall that L2loc(Emicro) is the space of all measurable functions f E rarr Csuch that for every bounded set B sub E we haveint

B

|f |2 dmicro lt +infin (23)

By choosing an exhausting family Bn of bounded sets (for example balls withfixed centre whose radii tend to infinity) and using (23) for B = Bn we endowthe space L2loc(Emicro) with a countable family of seminorms that makes it intoa complete separable metric space Of course the resulting topology is independentof the choice of the exhausting family of sets

Let H sub L2loc(Emicro) be a vector subspace When Eprime sub E is a Borel set such thatχEprimeH is a closed subspace of L2(Emicro) we denote by ΠEprime

the operator of orthogonalprojection onto the subspace χEprimeH sub L2(Emicro) We now fix a Borel subset E0 sub EInformally speaking E0 is the set where the particles may accumulate We imposethe following conditions on E0 and H

Assumption 2 (1) For every bounded Borel set B sub E the space χE0cupBH isa closed subspace of L2(Emicro)

(2) For every bounded Borel set B sub E E0 we have

ΠE0cupB isin I1loc(Emicro) χBΠE0cupBχB isin I1(Emicro)

(3) If ϕ isin H satisfies χE0ϕ = 0 then ϕ = 0

If a subspace H and the subset E0 are such that every function ϕ isin H withχE0ϕ = 0 must be the zero function then we say that H has the property of uniqueextension from E0

Theorem 211 Let E be a locally compact metric space and micro a σ-finite Borelmeasure on E If a subspace H sub L2loc(Emicro) and a Borel subset E0 sub E sat-isfy Assumption 2 then there is a σ-finite Borel measure B on Conf(E) with thefollowing properties

(1) B-almost every configuration has at most finitely many particles outside E0(2) For every (possibly empty) bounded Borel set B sub E E0 we have 0 lt

B(Conf(EE0 cupB)) lt +infin and

B|Conf(EE0cupB)


The conditions (1) and (2) determine the measure B uniquely up to multiplicationby a positive constant

We write B(HE0) for the one-dimensional cone of non-zero infinite determi-nantal measures induced by H and E0 and slightly abusing the notation writeB = B(HE0) for a representative of this cone


Remark If B is a bounded set then by definition

B(HE0) = B(HE0 cupB)

Remark If Eprime sub E is a Borel subset such that χE0cupEprime is a closed subspace ofL2(Emicro) and the operator ΠE0cupEprime

of orthogonal projection onto χE0cupEprimeH satisfies

ΠE0cupEprimeisin I1loc(Emicro) χEprimeΠE0cupEprime

χEprime isin I1(Emicro)

then exhausting Eprime by bounded sets we easily obtain from Theorem 211 that0 lt B(Conf(EE0 cup Eprime)) lt +infin and

B|Conf(EE0cupEprime)

B(Conf(EE0 cup Eprime))= PΠE0cupEprime

25 Change of variables for infinite determinantal measures Any homeo-morphism F E rarr E induces a homeomorphism of Conf(E) which will again bedenoted by F For every configuration X isin Conf(E) the particles of F (X) are ofthe form F (x) for all x isin X

We now assume that the measures Flowastmicro and micro are equivalent and let B = B(HE0)be an infinite determinantal measure We introduce the subspace

F lowastH =ϕ(F (x))

radicdFlowastmicro

dmicro ϕ isin H

The following proposition is easily obtained from the definitions

Proposition 212 The push-forward of the infinite determinantal measure

B = B(HE0)

is given byFlowastB = B(F lowastHF (E0))

26 Example infinite orthogonal polynomial ensembles Let ρ be a non-negative function on R not identically equal to zero Take N isin N and endow theset RN with the measure

prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)

If for all k = 0 2N minus 2 we haveint +infin

minusinfinxkρ(x) dx lt +infin

then the measure (24) is finite and after normalization yields a determinantalpoint process on Conf(R)

1138 A I Bufetov

Given a finite family of functions f1 fN on the real line we writespan(f1 fN ) for the vector space spanned by these functions For a generalfunction ρ we introduce a subspace H(ρ) sub L2loc(R Leb) by the formula

H(ρ) = span(radic

ρ(x) xradicρ(x) xNminus1

radicρ(x)

)

The following obvious proposition shows that (24) is an infinite determinantal mea-sure

Proposition 213 Let ρ be a non-negative continuous function on R and let(a b) sub R be a non-empty interval such that the restriction of ρ to (a b) is positiveand bounded Then the measure (24) is an infinite determinantal measure of theform B(H(ρ) (a b))

27 Infinite determinantal measures and multiplicative functionals Ournext aim is to show that under some additional assumptions every infinite determi-nantal measure can be represented as the product of a finite determinantal measureand a multiplicative functional

Proposition 214 Let B = B(HE0) be the infinite determinantal measure inducedby a subspace H sub L2loc(Emicro) and a Borel set E0 let g E rarr (0 1] be a positiveBorel function such that

radicg H is a closed subspace of L2(Emicro) and let Πg be the

corresponding projection operator Assume further that(1)

radic1minus gΠE0

radic1minus g isin I1(Emicro)

(2) χEE0ΠgχEE0I1(Emicro)

(3) Πg isin I1loc(Emicro)Then the multiplicative functional Ψg is B-almost surely positive and B-integrableand we have

ΨgBintConf(E)

Ψg dB= PΠg

The proof uses several auxiliary propositionsFirst we have the following simple corollary of the unique extension property

Proposition 215 Suppose that H sub L2loc(Emicro) has the property of uniqueextension from E0 and let ψ isin L2loc(Emicro) be such that χE0cupBψ isin χE0cupBH forevery bounded Borel set B sub E E0 Then ψ isin H

Proof Indeed for every B there is a function ψB isin L2loc(Emicro) such thatχE0cupBψB =χE0cupBψ Take bounded Borel sets B1 and B2 and note that χE0ψB1 =χE0ψB2 = χE0ψ whence the unique extension property yields that ψB1 = ψB2 Therefore all the functions ψB are equal to each other and to ψ which thus belongsto H

Our next proposition gives a sufficient condition for a subspace of locally square-integrable functions to be closed in L2

Proposition 216 Let L sub L2loc(Emicro) be a subspace with the following proper-ties

(1) For every bounded Borel set B sub E E0 the subspace χE0cupBL is closedin L2(Emicro)


(2) The natural restriction map χE0cupBL rarr χE0L is an isomorphism of Hilbertspaces and the norm of its inverse is bounded above by a positive constant inde-pendent of B

Then L is a closed subspace of L2(Emicro) and the natural restriction map L rarrχE0L is an isomorphism of Hilbert spaces

Proof If L contains a function with non-integrable square then the inverse of therestriction isomorphism χE0cupBLrarr χE0L will have an arbitrarily large norm for anappropriate choice of B Hence it follows from the unique extension property andProposition 215 that L is closed

We now proceed to prove Proposition 214We first check that the following inclusion holds for every bounded Borel set

B sub E E0 radic1minus gΠE0cupB


Indeed by the definition of an infinite determinantal measure we have

χBΠE0cupB isin I2(Emicro)

whence a fortiori radic1minus g χBΠE0cupB isin I2(Emicro)

We now recall that

ΠE0 = χE0ΠE0cupB(1minus χBΠE0cupB)minus1ΠE0cupBχE0

Then the relation radic1minus gΠE0


implies that radic1minus g χE0Π

E0cupBχE0


or equivalently radic1minus g χE0Π

E0cupB isin I2(Emicro)

We conclude that radic1minus gΠE0cupB isin I2(Emicro)

or in other words radic1minus gΠE0cupB


as requiredWe now check that the subspace

radicg HχE0cupB is closed in L2(Emicro) This follows

directly from the closure ofradicg H the property of unique extension from E0 (the

subspaceradicg H possesses it since H does) and the assumption χEE0Π

gχEE0 isinI1(Emicro)

Let ΠgχE0cupB be the operator of orthogonal projection ontoradicg HχE0cupB

It follows from the above that for every bounded Borel set B sub E E0 themultiplicative functional Ψg is PΠE0cupB -almost surely positive and furthermore

ΨgPΠE0cupBintΨg dPΠE0cupB

= PΠgχE0cupB

1140 A I Bufetov

Therefore for every bounded Borel set B sub E E0 we have

ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)

It remains to note that the assertion of Proposition 214 follows immediatelyfrom (25)

28 Infinite determinantal measures obtained as finite-rank perturba-tions of probability determinantal measures

281 Construction of finite-rank perturbations We now consider infinite determi-nantal measures induced by those subspaces H that are obtained by adding a finite-dimensional subspace V to a closed subspace L sub L2(Emicro)

Let Q isin I1loc(Emicro) be the operator of orthogonal projection onto the closedsubspace L sub L2(Emicro) and let V a finite-dimensional subspace of L2loc(Emicro) withV cap L2(Emicro) = 0 We put H = L+ V Let E0 sub E be a Borel subset We imposethe following restrictions on L V and E0

Assumption 3 (1) χEE0QχEE0 isin I1(Emicro)(2) χE0V sub L2(Emicro)(3) if ϕ isin V satisfies χE0ϕ isin χE0L then ϕ = 0(4) if ϕ isin L satisfies χE0ϕ = 0 then ϕ = 0

Proposition 217 If L V and E0 satisfy Assumption 3 then the subspace H =L+ V and the set E0 satisfy Assumption 2

In particular for every bounded Borel set B the subspace χE0cupBL is closedThis can be seen by taking Eprime = E0 cupB in the following obvious proposition

Proposition 218 Let Q isin I1loc(Emicro) be the operator of orthogonal projectiononto a closed subspace L sub L2(Emicro) Let Eprime sub E be a Borel subset such thatχEprimeQχEprime isin I1(Emicro) and for every function ϕ isin L the equality χEprimeϕ = 0 impliesthat ϕ = 0 Then the subspace χEprimeL is closed in L2(Emicro)

Thus the subspaceH and the Borel subset E0 determine an infinite determinantalmeasure B = B(HE0) The measure B(HE0) is infinite by Proposition 210

282 Multiplicative functionals of finite-rank perturbations Proposition 214 hasthe following immediate corollary

Corollary 219 Suppose that L V and E0 induce an infinite determinantal mea-sure B Let g E rarr (0 1] be a positive measurable function with the followingproperties

(1)radicg V sub L2(Emicro)

(2)radic

1minus gΠradic

1minus g isin I1(Emicro)Then the multiplicative functional Ψg is B-almost surely positive and B-integrableand we have

ΨgBintΨg dB

= PΠg

where Πg is the operator of orthogonal projection onto the closed subspaceradicg L+radic

g V


29 Example the infinite Bessel point process We are now ready to proveProposition 11 on the existence of the infinite Bessel point process B(s) s 6 minus1To do this we need the following property of the ordinary Bessel point process Jss gt minus1 As above we denote the range of the projection operator Js by Ls

Lemma 220 Take an arbitrary s gt minus1 Then the following assertions hold(1) For every R gt 0 the subspace χ(R+infin)Ls is closed in L2((0+infin) Leb) and

the corresponding projection operator JsR is locally of trace class(2) For every R gt 0 we have

PJs

(Conf((0+infin) (R+infin))

)gt 0

PJs|Conf((0+infin)(R+infin))

PJs


) = PJsR

Proof Clearly for every R gt 0 we haveint R

0

Js(x x) dx lt +infin

or equivalentlyχ(0R)Jsχ(0R) isin I1((0+infin) Leb)

The lemma now follows from the unique extension property of the Bessel pointprocess

We now take s 6 minus1 and recall that the number ns isin N is defined by the relation

s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)

Proposition 221 We have dim V (s) = ns and for every R gt 0

χ(0R)V(s) cap L2((0+infin) Leb) = 0

Proof The following argument was suggested by Yanqi Qiu By definition of theBessel kernel every function in Ls+2ns is the restriction to R+ of a harmonic func-tion defined on the half-plane z Re(z) gt 0 The desired claim now follows fromthe uniqueness theorem for harmonic functions

Proposition 221 immediately yields the existence of the infinite Bessel pointprocess B(s) which completes the proof of Proposition 11

By making the change of variable y = 4x we establish the existence of themodified infinite Bessel point process B(s) Moreover using the characterization(described in Proposition 214 and Corollary 219) of multiplicative functionalsof infinite determinantal measures we arrive at the proof of Propositions 15ndash17

1142 A I Bufetov

Appendix A The Jacobi orthogonal polynomial ensemble

A1 Jacobi polynomials For α β gt minus1 let P (αβ)n be the standard Jacobi

polynomials namely polynomials on the closed interval [minus1 1] that are orthogonalwith the weight

(1minus u)α(1 + u)β

and normalized by the condition

P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)

We recall that the leading coefficient k(αβ)n of the polynomial P (αβ)

n is given by thefollowing formula (see for example [32] formula (4216))

k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)

and for the squared norm we have

h(αβ)n =

int 1

minus1

(P (αβ)n (u))2(1minus u)α(1 + u)β du

=2α+β+1

2n+ α+ β + 1Γ(n+ α+ 1)Γ(n+ β + 1)Γ(n+ 1)Γ(n+ α+ β + 1)

Let K(αβ)n (u1 u2) be the nth ChristoffelndashDarboux kernel of the Jacobi orthogonal

polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l

(1minusu1)α2(1+u1)β2(1minusu2)α2(1+u2)β2

The ChristoffelmdashDarboux formula gives an equivalent representation for the ker-nel K(αβ)

n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2

times (1minus u2)α2(1 + u2)β2P(αβ)n (u1)P

(αβ)nminus1 (u2)minus P

(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)

A2 The recurrence relations between Jacobi polynomials We havethe following recurrence relation between the ChristoffelndashDarboux kernels K(αβ)

n+1

and K(α+2β)n

Proposition A1 For all α β gt minus1

K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2

times P (α+1β)n (u2)(1minus u2)α2(1 + u2)β2 + K(α+2β)

n (u1 u2) (27)


Remark Taking the scaling limit of (27) we obtain a similar recurrence relationfor Bessel kernels the Bessel kernel with parameter s is a rank-one perturbation ofthe Bessel kernel with parameter s+2 This can also be easily established directlyUsing the recurrence relation

Js+1(x) =2sxJs(x)minus Jsminus1(x)

for the Bessel functions we immediately obtain the desired relation

Js(x y) = Js+2(x y) +s+ 1radicxy

Js+1(radicx)Js+1(

radicy)

for the Bessel kernels

Proof of Proposition A1 This routine calculation is included for completenessWe use the standard recurrence relations for Jacobi polynomials First we usethe relation(

n+α+ β

2+ 1

)(uminus 1)P (α+1β)

n (u) = (n+ 1)P (αβ)n+1 (u)minus (n+ α+ 1)P (αβ)

n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)

times (u1 minus 1)P (α+1β)n (u1)P

(αβ)n (u2)minus (u2 minus 1)P (α+1β)

n (u2)P(αβ)n (u1)

u1 minus u2 (28)

Then we use the relation

(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)

n (u)minus (n+ β)P (α+1β)nminus1 (u)

to arrive at the equality

(u1 minus 1)P (α+1β)n (u1)P


n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P

(α+1β)nminus1 (u2)minus (1minus u2)P

(α+1β)n (u2)P

(α+1β)nminus1 (u1)

u1 minus u2

(29)

Next using the recurrence relation(n+

α+ β + 12

)(1minus u)P (α+2β)

nminus1 (u) = (n+ α+ 1)P (α+1β)nminus1 (u)minus nP (α+1β)

n (u)

1144 A I Bufetov

we arrive at the equality

(1minus u1)P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)

(1minus u1)(1minus u2)

timesP

(α+1β)n (u1)P

(α+2β)nminus1 (u2)minus P

(α+1β)n (u2)P


u1 minus u2 (30)

Combining (29) and (30) we obtain



n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)

times (1minus u1)(1minus u2)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(31)Using the recurrence relation

(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)


we now arrive at the equality

P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)

Combining (28) (31) (32) and using the definition (26) of the ChristoffelndashDarbouxkernels we conclude the proof of Proposition A1

As above given a finite family of functions f1 fN on the interval [minus1 1] oron the real line we write span(f1 fN ) for the vector space spanned by thesefunctions For α β isin R we introduce the subspaces

L(αβn)Jac = span

((1minus u)α2(1 + u)β2 (1minus u)α2(1 + u)β2u

(1minus u)α2(1 + u)β2unminus1)

Proposition A1 yields the following orthogonal direct sum decomposition forα β gt minus1

L(αβn)Jac = CP (α+1β)

n oplus L(α+2βnminus1)Jac (33)

The relation (33) remains valid for α isin (minus2minus1] although the corresponding spacesare no longer subspaces of L2 It is convenient to restate (33) shifting α by 2


Proposition A2 For all α gt 0 β gt minus1 n isin N we have

L(αminus2βn)Jac = CP (αminus1β)

n oplus L(αβnminus1)Jac

Proof Let Q(αβ)n be functions of the second kind corresponding to the Jacobi poly-

nomials P (αβ)n By formula (46219) in [32] for all u isin (minus1 1) and ν gt 1 we have

nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12

(ν minus 1)minusα(ν + 1)minusβ

(ν minus u)+

2minusαminusβ

2n+ α+ β + 2

times Γ(n+ 2)Γ(n+ α+ β + 2)Γ(n+ α+ 1)Γ(n+ β + 1)

P(αβ)n+1 (u)Q(αβ)

n (ν)minusQ(αβ)n+1 (ν)P (αβ)

n (u)ν minus u

We pass to the limit as ν rarr 1 using the following asymptotic expansion as ν rarr 1for Jacobi functions of the second kind (see [32] formula (4625))

Q(α)n (v) sim 2αminus1Γ(α)Γ(n+ β + 1)

Γ(n+ α+ β + 1)(ν minus 1)minusα

Recalling the recurrence formula (22719) in [33]

P(αminus1β)n+1 (u) = (n+ α+ β + 1)P (αβ)

n+1 minus (n+ β + 1)P (αβ)n (u)

we arrive at the relation

11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)

P(αminus1β)n+1 isin L(αβn)

Jac

which immediately yields Proposition A2

We now take s gt minus1 and for brevity write P (s)n = P

(s0)n

The leading coefficient k(s)n and the squared norm h

(s)n of the polynomial P (s)

n

are given by the formulae

k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1

(P (s)n (u))2(1minus u)s du =

2s+1

2n+ s+ 1

We denote the corresponding nth ChristoffelndashDarboux kernel by K(s)n (u1 u2)

K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l

(1minus u1)s2(1minus u2)s2 (34)

1146 A I Bufetov

The ChristoffelndashDarboux formula gives an equivalent representation of the ker-nel K(s)

n

K(s)n (u1 u2)

=n(n+ s)

2s(2n+ s)(1minus u1)s2(1minus u2)s2P

(s)n (u1)P

(s)nminus1(u2)minus P

(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)

A3 The Bessel kernel Consider the half-line (0+infin) with the standardLebesgue measure Leb Take s gt minus1 and consider the standard Bessel kernel

Js(y1 y2) =radicy1Js+1(

radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)

(see for example [5] p 295)An alternative integral representation for the kernel Js is given by

Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)

(see for example formula (22) on p 295 in [5])It follows from (36) that the kernel Js induces on L2((0+infin) Leb) an operator

of orthogonal projection onto the subspace of functions whose Hankel transformvanishes everywhere outside [0 1] (see [5])

Proposition A3 For every s gt minus1 as nrarrinfin the kernel K(s)n converges to Js

uniformly in all variables on compact subsets of (0+infin)times (0+infin)

Proof This follows immediately from the classical HeinendashMehler asymptotics forJacobi orthogonal polynomials (see for example [32] Ch VIII) Note that theuniform convergence actually holds on arbitrary simply connected compact subsetsof (C 0)times (C 0)

Appendix B Spaces of configurationsand determinantal point processes

B1 Spaces of configurations Let E be a locally compact complete metricspace

A configuration X on E is an unordered family of points called particles Themain assumption is that the particles do not accumulate anywhere in E or equiva-lently that every bounded subset of E contains only finitely many particles of theconfiguration

With each configuration X we associate a Radon measuresumxisinX

δx

where the sum is taken over all particles in X Conversely every purely atomicRadon measure on E is given by a configuration Thus the space Conf(E) of all


configurations on E can be identified with a closed subset in the set of all integer-valued Radon measures on E This enables us to endow Conf(E) with the structureof a complete metric space which is however not locally compact

The Borel structure on Conf(E) can be defined equivalently as follows For everybounded Borel subset B sub E we introduce the function

B Conf(E) rarr R

sending every configuration X to the number of its particles lying in B The familyof functions B over all bounded Borel subsets B of E determines a Borel struc-ture on Conf(E) In particular to define a probability measure on Conf(E) it isnecessary and sufficient to determine the joint distributions of the random variablesB1 Bk

for all finite tuples of disjoint bounded Borel subsets B1 Bk sub E

B2 The weak topology on the space of probability measures on thespace of configurations The space Conf(E) is endowed with the natural struc-ture of a complete metric space and therefore the space Mfin(Conf(E)) of finiteBorel measures on the space of configurations is also a complete metric space withrespect to the weak topology

Let ϕ E rarr R be a compactly supported continuous function We define a mea-surable function ϕ Conf(E) rarr R by the formula

ϕ(X) =sumxisinX

ϕ(x)

For a bounded Borel set B sub E we have B = χB

The Borel σ-algebra on Conf(E) coincides with the σ-algebra generated by theinteger-valued random variables B over all bounded Borel subsets B sub E There-fore it also coincides with the σ-algebra generated by the random variables ϕ

over all compactly supported continuous functions ϕ E rarr R Thus the followingproposition holds

Proposition B1 Every Borel probability measure P isin Mfin(Conf(E)) is uniquelydetermined by the joint distributions of the random variables

ϕ1 ϕ2 ϕl

over all possible finite tuples of continuous functions ϕ1 ϕl E rarr R with dis-joint compact supports

The weak topology on Mfin(Conf(E)) admits the following characterization interms of these finite-dimensional distributions (see [34] vol 2 Theorem 111VII)Let Pn n isin N and P be Borel probability measures on Conf(E) Then the mea-sures Pn converge weakly to P as n rarr infin if and only if for every finite tupleϕ1 ϕl of continuous functions with disjoint compact supports the joint distri-butions of the random variables ϕ1 ϕl

with respect to Pn converge as nrarrinfinto the joint distribution of ϕ1 ϕl

with respect to P (the convergence of jointdistributions is understood in the sense of the weak topology on the space of Borelprobability measures on Rl)

1148 A I Bufetov

B3 Spaces of locally trace-class operators Let micro be a σ-finite Borelmeasure on E We write I1(Emicro) for the ideal of all trace-class operators K L2(Emicro) rarr L2(Emicro) (a precise definition is given for example in vol 1 of [35]and in [36]) and denote the I1-norm of an operator K by KI1 We also writeI2(Emicro) for the ideal of HilbertndashSchmidt operators K L2(Emicro) rarr L2(Emicro) anddenote the I2-norm of K by KI2

Let I1loc(Emicro) be the space of operators K L2(Emicro) rarr L2(Emicro) such that forevery bounded Borel set B sub E we have

χBKχB isin I1(Emicro)

We endow the space I1loc(Emicro) with a countable family of seminorms

χBKχBI1

where as above B ranges over an exhausting family Bn of bounded sets

B4 Determinantal point processes A Borel probability measure P onConf(E) is said to be determinantal if there is an operator K isin I1loc(Emicro) suchthat the following equality holds for every bounded measurable function g withg minus 1 supported on a bounded set B

EPΨg = det(1 + (g minus 1)KχB) (37)

The Fredholm determinant in (37) is well defined since K isin I1loc(Emicro) The equa-tion (37) determines the measure P uniquely For every tuple of disjoint boundedBorel sets B1 Bl sub E and all z1 zl isin C it follows from (37) that

EPzB11 middot middot middot zBl

l = det(

1 +lsum

j=1

(zj minus 1)χBjKχF

i Bi

)

For further results and background on determinantal point processes see forexample [7] [24] [31] [37]ndash[43]

For every operator K in I1loc(Emicro) we denote the corresponding determinan-tal measure throughout by PK Note that PK is uniquely determined by K butdifferent operators may yield the same measure By the MacchindashSoshnikov theo-rem (see [44] [31]) every Hermitian positive contraction belonging to I1loc(Emicro)induces a determinantal point process

B5 Change of variables Every homeomorphism F E rarr E induces a homeo-morphism of the space Conf(E) which by a slight abuse of notation will again bedenoted by F For every X isin Conf(E) the particles of the configuration F (X)are of the form F (x) x isin X Let micro be a σ-finite measure on E and let PK bethe determinantal measure induced by an operator K isin I1loc(Emicro) We define anoperator FlowastK by the formula FlowastK(f) = K(f F )

Assume that the measures Flowastmicro and micro are equivalent and consider the operator

KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro


Note that if K is self-adjoint then so is KF If K is given by a kernel K(x y) thenKF is given by the kernel

KF (x y) =

radicdFlowastmicro

dmicro(x)K(Fminus1x Fminus1y)

radicdFlowastmicro

dmicro(y)

The following proposition is obtained directly from the definitions

Proposition B2 The action of the homeomorphism F on the determinantal mea-sure PK is given by the formula

FlowastPK = PKF

Note that if K is the operator of orthogonal projection onto a closed subspaceL sub L2(Emicro) then by definition KF is the operator of orthogonal projection ontothe closed subspace

ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)

B6 Multiplicative functionals on spaces of configurations Let g be anarbitrary non-negative measurable function on E We introduce the multiplicativefunctional Ψg Conf(E) rarr R by the formula

Ψg(X) =prodxisinX

g(x)

If the infinite productprod

xisinX g(x) converges absolutely to 0 or infin then we putΨg(X) = 0 or Ψg(X) = infin respectively If the product on the right-hand sidedoes not converge absolutely then the multiplicative functional is not definedat the point considered

B7 Determinantal point processes and multiplicative functionals Theconstruction of infinite determinantal measures is based on the results of [8] [9]which may informally be stated as follows the product of a determinantal mea-sure and a multiplicative functional is again a determinantal measure In otherwords if PK is the determinantal measure on Conf(E) induced by an operator Kon L2(Emicro) then it was shown under certain additional assumptions in [8] [9] thatthe measure ΨgPK yields a determinantal point process after normalization

As above let g be a non-negative measurable function on E If the operator1 + (g minus 1)K is invertible then we put

B(gK) = gK(1 + (g minus 1)K)minus1 B(gK) =radicg K(1 + (g minus 1)K)minus1radicg

By definition B(gK) B(gK) isin I1loc(Emicro) since K isin I1loc(Emicro) If K is self-adjoint then so is B(gK)

We now recall some propositions from [9]

1150 A I Bufetov

Proposition B3 Let K isin I1loc(Emicro) be a positive self-adjoint contractionPK the corresponding determinantal measure on Conf(E) and g a non-negativebounded measurable function on E such thatradic

g minus 1Kradicg minus 1 isin I1(Emicro) (38)

and the operator 1 + (g minus 1)K is invertible Then the following assertions hold(1) We have Ψg isin L1(Conf(E)PK) andint

Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0

(2) The operators B(gK) B(gK) induce a determinantal measure PB(gK) =P eB(gK) on Conf(E) satisfying

PB(gK) =ΨgPKint

Conf(E)Ψg dPK

Remark Suppose that the condition (38) holds and K is self-adjoint Then theoperator 1 + (g minus 1)K is invertible if and only if 1 +

radicg minus 1K

radicg minus 1 is

If Q is a projection operator then the operator B(gQ) admits the followingdescription

Proposition B4 Let L sub L2(Emicro) be a closed subspace Q the operator of orthog-onal projection onto L and g a bounded measurable non-negative function such thatthe operator 1 + (gminus 1)Q is invertible Then B(gQ) is the operator of orthogonalprojection onto the closure of

radicg L

We now consider the particular case when g is the characteristic function ofa Borel subset If Eprime sub E is a Borel subset such that the subspace χEprimeL is closed(we recall that Proposition 218 gives a sufficient condition for this) then we writeQEprime

for the operator of orthogonal projection onto χEprimeLWe obtain the following corollary of Proposition B3

Corollary B5 Let Q isin I1loc(Emicro) be the operator of orthogonal projection ontoa closed subspace L isin L2(Emicro) and let Eprime sub E be a Borel subset such thatχEprimeQχEprime isin I1(Emicro) Then

PQ(Conf(EEprime)) = det(1minus χEEprimeQχEEprime)

Assume further that for every function ϕ isin L the equality χEprimeϕ = 0 implies thatϕ = 0 Then the subspace χEprimeL is closed and we have

PQ(Conf(EEprime)) gt 0 QEprimeisin I1loc(Emicro)

andPQ|Conf(EEprime)

PQ(Conf(EEprime))= PQEprime


Thus the measure induced from a determinantal measure on the set of config-urations all of whose particles lie in Eprime is again a determinantal measure In thecase of a discrete phase space the related induced processes were considered byLyons [39] and Borodin and Rains [45]

We now state a sufficient condition for a multiplicative functional to be positiveon almost all configurations

Proposition B6 If

micro(x isin E g(x) = 0) = 0radic|g minus 1|K

radic|g minus 1| isin I1(Emicro)

then0 lt Ψg(X) lt +infin

for PK-almost all configurations X isin Conf(E)

Proof Our assumptions imply that for PK-almost all X isin Conf(E) we havesumxisinX

|g(x)minus 1| lt +infin

which is in its turn sufficient for the absolute convergence of the infinite productprodxisinX g(x) to a finite non-zero limit

We state a version of Proposition B3 in the particular case when the function gassumes no values less than 1 In this case the multiplicative functional Ψg isautomatically non-zero and we obtain the following result

Proposition B7 Let Π isin I1loc(Emicro) be the operator of orthogonal projectiononto a closed subspace H and let g be a bounded Borel function on E such thatg(x) gt 1 for all x isin E Assume thatradic

g minus 1 Πradicg minus 1 isin I1(Emicro)

Then the following assertions hold(1) We have Ψg isin L1(Conf(E)PΠ) andint

Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)

(2) We haveΨgPΠintΨg dPΠ

= PΠg

where Πg is the operator of orthogonal projection ontoradicg H

Appendix C Construction of Pickrell measuresand proof of Proposition 18

We recall that the Pickrell measures are naturally defined on the space of mtimesnrectangular matrices

1152 A I Bufetov

Let Mat(mtimes nC) be the space of complex mtimes n matrices

Mat(mtimes nC) =z = (zij) i = 1 m j = 1 n

We write dz for the Lebesgue measure on Mat(mtimes nC)Take s isin R and m0 n0 isin N such that m0 +s gt 0 n0 +s gt 0 Following Pickrell

we take m gt m0 n gt n0 and introduce a measure micro(s)mn on Mat(mtimes nC) by the

formulamicro(s)

mn = const(s)mn det(1 + zlowastz)minusmminusnminuss dz

where

const(s)mn = πminusmnmprod

l=m0

Γ(l + s)Γ(n+ l + s)

For m1 6 m and n1 6 n we consider the natural projections

πmnm1n1

Mat(mtimes nC) rarr Mat(m1 times n1C)

Proposition C1 Let mn isin N be such that s gt minusm minus 1 Then for all z isinMat(nC) we haveint

(πm+1nmn )minus1(z)

det(1+zlowastz)minusmminusnminus1minuss dz = πn Γ(m+ 1 + s)Γ(n+m+ 1 + s)

det(1+zlowastz)minusmminusnminuss

Proposition 18 is an immediate corollary of Proposition C1

Proof of Proposition C1 As already mentioned in the introduction the followingcomputation goes back to the classical work of Hua Loo-Keng [10] Take z isinMat((m+ 1)timesnC) Multiplying if necessary by a unitary matrix on the left andon the right we represent the matrix πm+1n

mn z = z in diagonal form with positivereal diagonal entries zii = ui gt 0 i = 1 n zij = 0 for i 6= j

Here we put ui = 0 when i gt min(nm) Writing ξi = zm+1i for i = 1 nwe transform the determinant as follows

det(1 + zlowastz)minusmminus1minusnminuss =mprod

i=1

(1 + u2i )minusmminus1minusnminuss

(1 + ξlowastξ minus

nsumi=1

|ξi|2 u2i

1 + u2i

)minusmminus1minusnminuss

Simplify the expression in parentheses

1 + ξlowastξ minusnsum

i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i

Integrating with respect to ξ we obtainint (1+

nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0

rnminus1(1+ r)minusmminus1minusnminuss dr

where

r = r(m+1n)(z) =nsum

i=1

|ξi|2

1 + u2i

(39)


Using the Euler integralint +infin

0

rnminus1(1 + r)minusmminus1minusnminuss dr =Γ(n)Γ(m+ 1 + s)Γ(n+ 1 +m+ s)

we arrive at the desired equality

We introduce a map

πm+1nmn Mat((m+ 1)times nC) rarr Mat(mtimes nC)times R+

by the formulaπm+1n

mn (z) = (πm+1nmn (z) r(m+1n)(z))

where r(m+1n)(z) is given by the formula (39) Let P (mns) be a probability mea-sure on R+ such that

dP (mns)(r) =Γ(n+m+ s)Γ(n)Γ(m+ s)

rnminus1(1minus r)minusmminusnminuss dr

The measure P (mns) is well defined for m+ s gt 0

Corollary C2 For all mn isin N and s gt minusmminus 1 we have(πm+1n

mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)

Indeed this is precisely what was shown by our computationsRemoving a column is similar to removing a row(

πmn+1mn (z)

)t = πm+1nmn (zt)

We introduce the notation r(mn+1)(z) = r(n+1m)(zt) and define a map

πmn+1mn Mat(mtimes (n+ 1)C) rarr Mat(mtimes nC)times R+

by the formulaπmn+1

mn (z) =(πmn+1

mn (z) r(mn+1)(z))

Corollary C3 For all mn isin N and s gt minusmminus 1 we have(πmn+1

mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)

We now take an n such that n+ s gt 0 and define the map

πn Mat(Ntimes NC) rarr Mat(ntimes nC)

by the formula

πn(z) =(πinfininfin

nn (z) r(n+1n) r (n+1n+1) r(n+2n+1) r (n+2n+2) )

1154 A I Bufetov

Recalling the definition of the Pickrell measure micro(s) on Mat(NtimesNC) (see sect 171)we can now restate the result of our computations as follows

Proposition C4 If n+ s gt 0 then

(πn)lowastmicro(s) = micro(s)nn times

infinprodl=0

(P (n+l+1n+ls) times P (n+l+1n+l+1s)

) (40)

Bibliography

[1] D Pickrell ldquoMeasures on infinite dimensional Grassmann manifoldsrdquo J FunctAnal 702 (1987) 323ndash356

[2] D M Pickrell ldquoMackey analysis of infinite classical motion groupsrdquo PacificJ Math 1501 (1991) 139ndash166

[3] G Olrsquoshanskii and A Vershik ldquoErgodic unitarily invariant measures on the spaceof infinite Hermitian matricesrdquo Contemporary mathematical physics Amer MathSoc Transl Ser 2 vol 175 Amer Math Soc Providence RI 1996 pp 137ndash175

[4] A Borodin and G Olshanski ldquoInfinite random matrices and ergodic measuresrdquoComm Math Phys 2231 (2001) 87ndash123

[5] C A Tracy and H Widom ldquoLevel spacing distributions and the Bessel kernelrdquoComm Math Phys 1612 (1994) 289ndash309

[6] A I Bufetov ldquoFiniteness of ergodic unitarily invariant measures on spaces ofinfinite matricesrdquo Ann Inst Fourier (Grenoble) 643 (2014) 893ndash907 arXiv11082737

[7] T Shirai and Y Takahashi ldquoRandom point fields associated with certain Fredholmdeterminants II Fermion shifts and their ergodic and Gibbs propertiesrdquo AnnProbab 313 (2003) 1533ndash1564

[8] A I Bufetov ldquoMultiplicative functionals of determinantal processesrdquo Uspekhi MatNauk 671(403) (2012) 177ndash178 English transl Russian Math Surveys 671(2012) 181ndash182

[9] A I Bufetov ldquoInfinite determinantal measuresrdquo Electron Res Announc MathSci 20 (2013) 12ndash30

[10] L K Hua Harmonic analysis of functions of several complex variables in theclassical domains translated from the Chinese (Science Press Peking 1958) InostrLit Moscow 1959 (Russian) English transl Transl Math Monogr vol 6 AmerMath Soc Providence RI 1963

[11] Yu A Neretin ldquoHua-type integrals over unitary groups and over projective limitsof unitary groupsrdquo Duke Math J 1142 (2002) 239ndash266

[12] P Bourgade A Nikeghbali and A Rouault ldquoEwens measures on compact groupsand hypergeometric kernelsrdquo Seminaire de Probabilites XLIII Lecture Notesin Math vol 2006 Springer Berlin 2011 pp 351ndash377

[13] A I Bufetov ldquoInfinite determinantal measures and the ergodic decomposition ofinfinite Pickrell measures IIrdquo Izv Ross Akad Nauk Ser Mat (to appear)

[14] A I Bufetov ldquoInfinite determinantal measures and the ergodic decomposition ofinfinite Pickrell measures IIIrdquo Izv Ross Akad Nauk Ser Mat (to appear)

[15] A Kolmogoroff Grundbegriffe der Wahrscheinlichkeitsrechnung Ergeb MathGrenzgeb vol 2 J Springer Berlin 1933 Russian transl 2nd ed NaukaMoscow 1974


[16] G I Olshanskii ldquoUnitary representations of infinite dimensional pairs (G K) andthe formalism of R Howerdquo Representation of Lie groups and related topics AdvStud Contemp Math vol 7 Gordon and Breach New York 1990 pp 269ndash463httpwwwiitpruuploaduserpage52HoweFormpdf

[17] G I Olrsquoshanskii Unitary representations of infinite-dimensional classical groupsDiss Doct Phys-Math Sci Institute of Geography Academy of Sciences of theUSSR Moscow 1989 iitpruuploaduserpage52Olshanski thesispdf (Russian)

[18] A I Bufetov ldquoErgodic decomposition for measures quasi-invariant under a Borelaction of an inductively compact grouprdquo Mat Sb 2052 (2014) 39ndash70 Englishtransl Sb Math 2052 (2014) 192ndash219 Ergodic decomposition for measuresquasi-invariant under Borel actions of inductively compact groups arXiv 11050664

[19] D Pickrell ldquoSeparable representations of automorphism groups of infinitesymmetric spacesrdquo J Funct Anal 901 (1990) 1ndash26

[20] A M Vershik ldquoDescription of invariant measures for the actions of some infinite-dimensional groupsrdquo Dokl Akad Nauk SSSR 2184 (1974) 749ndash752 Englishtransl Soviet Math Dokl 155 (1974) 1396ndash1400

[21] M Rabaoui ldquoAsymptotic harmonic analysis on the space of square complexmatricesrdquo J Lie Theory 183 (2008) 645ndash670

[22] M Rabaoui ldquoA Bochner type theorem for inductive limits of Gelfand pairsrdquo AnnInst Fourier (Grenoble) 585 (2008) 1551ndash1573

[23] V I Bogachev Measure theory vol II Springer-Verlag Berlin 2007

[24] T Shirai and Y Takahashi ldquoRandom point fields associated with certain Fredholmdeterminants I Fermion Poisson and boson point processesrdquo J Funct Anal2052 (2003) 414ndash463

[25] J Faraut Analyse sur les groupes de Lie Une introduction Calvage et MounetParis 2006

[26] J Faraut Analysis on Lie groups Cambridge Stud Adv Math vol 110Cambridge Univ Press Cambridge 2008

[27] S Ghobber and P Jaming ldquoStrong annihilating pairs for the FourierndashBesseltransformrdquo J Math Anal Appl 3772 (2011) 501ndash515

[28] S Ghobber and P Jaming ldquoUncertainty principles for integral operatorsrdquo StudiaMath 2203 (2014) 197ndash220 arXiv 12061195v1

[29] E M Baruch The classical Hankel transform in the Kirillov model arXiv10105184v1

[30] A Lenard ldquoStates of classical statistical mechanical systems of infinitely manyparticles Irdquo Arch Rational Mech Anal 593 (1975) 219ndash239

[31] A Soshnikov ldquoDeterminantal random point fieldsrdquo Uspekhi Mat Nauk 555(335)(2000) 107ndash160 English transl Russian Math Surveys 555 (2000) 923ndash975

[32] G Szego Orthogonal polynomials rev ed Amer Math Soc Colloq Publvol 23 Amer Math Soc Providence RI 1959 Russian transl FizmatlitMoscow 1962

[33] M Abramowitz and I A Stegun (eds) Handbook of mathematical functionsNational Bureau of Standards Applied Mathematics Series vol 55 Superintendentof Documents US Government Printing Office Washington DC 1964 Russiantransl Nauka Moscow 1979

[34] D J Daley and D Vere-Jones An introduction to the theory of point processesvols I II 2nd ed Probab Appl (NY) Springer-Verlag New York 2003 2008

[35] M Reed and B Simon Methods of modern mathematical physics vol I 2nd edAcademic Press New YorkndashLondon 1980 vols IIndashIV 1st ed 1975 1979 1978Russian transl vols 1ndash4 Mir Moscow 1977 1978 1982 1982

1156 A I Bufetov

[36] B Simon Trace ideals and their applications 2nd ed Math Surveys Monogrvol 120 Amer Math Soc Providence RI 2005

[37] A M Borodin ldquoDeterminantal point processesrdquo The Oxford handbook of randommatrix theory Oxford Univ Press Oxford 2011 pp 231ndash249

[38] J Ben Hough M Krishnapur Y Peres and B Virag ldquoDeterminantal processesand independencerdquo Probab Surv 3 (2006) 206ndash229

[39] R Lyons ldquoDeterminantal probability measuresrdquo Publ Math Inst Hautes EtudesSci 981 (2003) 167ndash212

[40] R Lyons and J E Steif ldquoStationary determinantal processes phase multiplicityBernoullicity entropy and dominationrdquo Duke Math J 1203 (2003) 515ndash575

[41] E Lytvynov ldquoFermion and boson random point processes as particle distributionsof infinite free Fermi and Bose gases of finite densityrdquo Rev Math Phys 1410(2002) 1073ndash1098

[42] T Shirai and Y Takahashi ldquoRandom point fields associated with fermion bosonand other statisticsrdquo Stochastic analysis on large scale interacting systems AdvStud Pure Math vol 39 Math Soc Japan Tokyo 2004 pp 345ndash354

[43] G Olshanski ldquoThe quasi-invariance property for the Gamma kernel determinantalmeasurerdquo Adv Math 2263 (2011) 2305ndash2350

[44] O Macchi ldquoThe coincidence approach to stochastic point processesrdquo Advances inAppl Probability 71 (1975) 83ndash122

[45] A Borodin and E M Rains ldquoEynardndashMehta theorem Schur process and theirPfaffian analogsrdquo J Stat Phys 1213-4 (2005) 291ndash317

Alexander I Bufetov

Steklov Mathematical Institute

Russian Academy of Sciences Moscow

Institute of information transmission problems


National Research University lsquoHigher

School of Economicsrsquo Moscow

Aix-Marseille Universite CNRS

Centrale Marseille I2M UMR 7373

E-mail bufetovmirasru

Received 6APR15Translated by P P NIKITIN

sect1 Introduction

11 Informal outline of main results

12 Historical remarks

13 Organization of the paper


141 Outline of the construction

142 The Jacobi orthogonal polynomial ensemble

143 The recurrence relation for Jacobi orthogonal polynomial ensembles

144 Scaling limits

145 Definition of the infinite Bessel point process

15 The modified Bessel point process

16 The modified infinite Bessel point process


171 Pickrell measures

18 Classification of ergodic measures

19 Statement of the main result

110 A skew-product representation of the measure $mathbb B^(s)$


1111 Approximation

1112 Convergence of probability measures on the Pickrell set

112 The radial part of a Pickrell measure

sect2 Construction and properties of infinite determinantal measures

21 Preliminary remarks on $sigma$-finite measures


221 Extension from a subset


23 Inductively determinantal measures

24 General construction of infinite determinantal measures

25 Change of variables for infinite determinantal measures

26 Example infinite orthogonal polynomial ensembles

27 Infinite determinantal measures and multiplicative functionals

28 Infinite determinantal measures obtained as finite-rank perturbations of probability determinantal measures

281 Construction of finite-rank perturbations

282 Multiplicative functionals of finite-rank perturbations

29 Example the infinite Bessel point process


A1 Jacobi polynomials

A2 The recurrence relations between Jacobi polynomials

A3 The Bessel kernel

Appendix B Spaces of configurations and determinantal point processes

B1 Spaces of configurations

B2 The weak topology on the space of probability measures on the space of configurations

B3 Spaces of locally trace-class operators

B4 Determinantal point processes

B5 Change of variables

B6 Multiplicative functionals on spaces of configurations

B7 Determinantal point processes and multiplicative functionals

Appendix C Construction of Pickrell measures and proof of Proposition 18

Bibliography

1112 A I Bufetov

If the constants constns are chosen appropriately then the Pickrell family ofmeasures has the Kolmogorov property of consistency with respect to natural pro-jections the push-forward of the Pickrell measure micro

(s)n+1 under the natural pro-

jection sending each (n + 1) times (n + 1) matrix to its upper left n times n corner isprecisely the Pickrell measure micro(s)

n This consistency property also holds for infinitePickrell measures provided that n is sufficiently large (see Proposition 18 for anexact statement) The consistency property and Kolmogorovrsquos theorem enable oneto define the Pickrell family of measures micro(s) s isin R on the space of infinite com-plex matrices The Pickrell measures are invariant under the actions of the infiniteunitary group on the left and right and the Pickrell family of measures is the nat-ural infinite-dimensional analogue of the canonical unitarily invariant measure ona Grassmann manifold (see the paper of Pickrell [1])

What is the ergodic decomposition of Pickrell measures with respect to the actionof the Cartesian square of the infinite unitary group The ergodic unitarily invari-ant probability measures on the space of infinite complex matrices were explicitlydescribed by Pickrell [2] and another approach to their description was given byVershik and Olshanski [3] Each ergodic measure is parametrized by an infinitearray x = (x1 xn ) of numbers in the half-line satisfying x1 gt x2 gt middot middot middot gt 0and x1 + middot middot middot + xn + middot middot middot lt +infin and an additional parameter γ that we call theGaussian parameter Informally speaking the parameters xn may be thought of asthe lsquoasymptotic eigenvaluesrsquo of an infinite complex matrix and γ as the differencebetween the lsquoasymptotic tracersquo and the sum of the asymptotic eigenvalues (thisdifference is positive in particular for a Gaussian measure)

In 2000 Borodin and Olshanski [4] proved that for finite Pickrell measures theGaussian parameter vanishes almost surely and the corresponding spectral measure(the measure parametrizing the ergodic decomposition) regarded as a measure onthe space of configurations on the half-line (0+infin) coincides with the Bessel pointprocess of Tracy and Widom [5] The correlation functions of this process areexpressed as determinants of the Bessel kernel

Borodin and Olshanski [4] posed the following problem Describe the ergodicdecomposition of infinite Pickrell measures In this paper we give a solution

The first step is the result in [6] saying that almost all ergodic components of aninfinite Pickrell measure are themselves finite only the spectral measure is infiniteThen it is proved as for finite measures that the Gaussian parameter vanishesalmost surely The ergodic decomposition is thus given by a σ-finite measure B(s)

on the space of configurations on the half-line (0+infin)How to describe σ-finite measures on the space of configurations We note that

the formalism of correlation functions is completely inapplicable since they aredefined only for finite measures

This paper gives for a first time an explicit method for constructing infinitemeasures on the space of configurations Since these measures are very closelyrelated to determinantal probability measures we call them infinite determinantalmeasures

We give three descriptions of the measure B(s) The first two may be carried outin much greater generality





Ψg(X) =prodxisinX

g(x)


ΨgB(s) (1)





Bn = E

1114 A I Bufetov


















Pn+1|Conf(EEn)



B|Conf(EEn)

B(Conf(EEn))= Pn


1116 A I Bufetov







(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)


ui = 1minus yi

2n2 i = 1 n (4)










constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography





Ψg(X) =prodxisinX

g(x)


ΨgB(s) (1)





Bn = E

1114 A I Bufetov


















Pn+1|Conf(EEn)



B|Conf(EEn)

B(Conf(EEn))= Pn


1116 A I Bufetov







(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)


ui = 1minus yi

2n2 i = 1 n (4)










constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1114 A I Bufetov


















Pn+1|Conf(EEn)



B|Conf(EEn)

B(Conf(EEn))= Pn


1116 A I Bufetov







(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)


ui = 1minus yi

2n2 i = 1 n (4)










constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography








Pn+1|Conf(EEn)



B|Conf(EEn)

B(Conf(EEn))= Pn


1116 A I Bufetov







(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)


ui = 1minus yi

2n2 i = 1 n (4)










constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1116 A I Bufetov







(ui minus uj)2nprod

i=1

(1minus ui)sdui (3)


ui = 1minus yi

2n2 i = 1 n (4)










constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography







constns

prod16iltj6n

(ui minus uj)2nprod

i=1

(1minus ui)sdui =1n

det K(s)n (ui uj)

nprodi=1

dui (5)



n


L(sn)Jac = span


)= span


)



(s+2nminus1)Jac


K(s)n (u1 u2) =

s+ 12s+1



n (u1 u2)


L(sn)Jac = CP (s+1)


1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1118 A I Bufetov


s

2+ ns isin

(minus1

212








Note thatL






limnrarrinfin



limnrarrinfin

2minusα+1


αminus12 =

Jα(radicyi)radicyi




radicy








(s) lies in L2









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography









)lt +infin










J (s)(x1 x2) =Js

(2radicx1

)1radicx2Js+1

(2radicx2

)minus Js

(2radicx2

)1radicx1Js+1

(2radicx1

)x1 minus x2

x1 gt 0 x2 gt 0

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1120 A I Bufetov

or equivalently

J (s)(x y) =1

x1x2

int 1

0

Js

(2radic

t

x1

)Js

(2radic

t

x2

)dt







L(s) =ϕ(4x)x

ϕ isin L(s)



V (s) =ϕ(4x)x

ϕ isin V (s)

H(s) =

ϕ(4x)x

ϕ isin H(s)




x






)lt +infin















S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography












S(X) =sumxisinX

x



exp(minusβS(X)

)isin L1


)




is determinantal


exp(minusβx

2

)H(s) (8)


((0+infin) Leb



1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1122 A I Bufetov








Mat(nC) =z = (zij) i = 1 n j = 1 n


πnn1



n1(z) = (zij) i j = 1 n1






projections πnn1


(πn+1n )minus1(z)








l=1

Γ(2l + s)Γ(2l minus 1 + s)(Γ(l + s))2

micro(s)n






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography






( nprodl=n0


)micro(s)








micro(s) =int

Merg(Mat(NC))

η dmicro(s)(η)


1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1124 A I Bufetov




n gt 0


(z(n))lowastz(n)


i = λ(n)i (z)





xi = limnrarrinfin

λ(n)i (z)n2

γ = limnrarrinfin

tr(z(n))lowastz(n)

n2 (10)






infinsumi=1

xi




Mat(NC)


(minus4

(γ minus

suminfink=1 xk

)λ2


(11)


Mat(NC)















on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography













on ΩP such that

micro(s) =int

ΩP




Ω0P =


infinsumn=1

xn



1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1126 A I Bufetov
























onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography



onto H(sR)


B(s) =int infin

0





micro(z max

16ij6n0|zij | lt R

)lt +infin



micro =int

ΩP




(λ(n)1 λ

(n)2 λ(n)

n )


z(n) = (zij)ij=1n



by the formula

r(n)(z) =(

1n2


(n)1

n2λ

(n)2

n2

λ(n)n

n2 0 0

)

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1128 A I Bufetov


r(n)(z) isin Ω0P










)= 0


by the formula


)











limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography



limnrarrinfin



limnrarrinfin

intMat(NC)


Mat(NC)



limnrarrinfin

intΩP


ΩP














S(ω) =infinsum

n=1

xn(ω)



1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1130 A I Bufetov







(1) ν(sβ)(ΩP Ω0P ) = 0










(λ1(z) λn(z)

) (15)









prodiltj






prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography





prodiltj

(λi minus λj)21

(1 + λi)2n+sdλ



(16)




prodiltj

(ui minus uj)2nprod

i=1










Y =infin⋃

n=1

Yn

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1132 A I Bufetov



B|Yn

B(Yn)= Pn (18)


PN |Yn

PN (Yn)= Pn




andB1|Yn

B1(Yn)=

B2|Yn

B2(Yn)











infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography




infxisinE0

g(x) gt 0 (20)







radicg (21)










0



1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1134 A I Bufetov

























trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography







trχEE0ΠχEE0 6 R















B(Conf(E E0 cupB))


1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1136 A I Bufetov



B













B|Conf(EE0cupB)

















radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography














radicdFlowastmicro

dmicro ϕ isin H



B = B(HE0)



prod16ij6N

(xi minus xj)2Nprod

i=1

ρ(xi) dxi (24)




1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1138 A I Bufetov


H(ρ) = span(radic


radicρ(x)

)







radic1minus gΠE0




ΨgBintConf(E)

Ψg dB= PΠg

















We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography











We now recall that





E0cupBχE0















= PΠgχE0cupB

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1140 A I Bufetov


ΨgχE0cupBBint

ΨgχE0cupBdB

= PΠgχE0cupB (25)













(2)radic

1minus gΠradic


ΨgBintΨg dB

= PΠg


g V





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography





PJs


)gt 0


PJs


) = PJsR


0





s

2+ ns isin

(minus1

212

]

Put


Js+2nsminus1

(radicy)

radicy

)






1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1142 A I Bufetov






P (αβ)n (1) =

Γ(n+ α+ 1)Γ(n+ 1)Γ(α+ 1)



k(αβ)n =

Γ(2n+ α+ β + 1)2nΓ(n+ 1)Γ(n+ α+ β + 1)


h(αβ)n =

int 1

minus1


=2α+β+1



polynomial ensemble

K(αβ)n (u1 u2) =

nminus1suml=0

P(αβ)l (u1)P

(αβ)l (u2)

h(αβ)l



n

K(αβ)n (u1 u2) =

2minusαminusβ

2n+ α+ β

Γ(n+ 1)Γ(n+ α+ β + 1)Γ(n+ α)Γ(n+ β)

(1minus u1)α2(1 + u1)β2



(αβ)n (u2)P

(αβ)nminus1 (u1)

u1 minus u2 (26)


n+1

and K(α+2β)n


K(αβ)n+1 (u1 u2)

=α+ 1

2α+β+1

Γ(n+ 1)Γ(n+ α+ β + 2)Γ(n+ β + 1)Γ(n+ α+ 1)

P (α+1β)n (u1)(1minus u1)α2(1 + u1)β2


n (u1 u2) (27)






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography






Js+1(radicx)Js+1(

radicy)



n+α+ β

2+ 1



n (u)

to obtain that

P(αβ)n+1 (u1)P

(αβ)n (u2)minus P

(αβ)n+1 (u2)P

(αβ)n (u1)

u1 minus u2=

2n+ α+ β + 22(n+ 1)



n (u2)P(αβ)n (u1)

u1 minus u2 (28)


(2n+ α+ β + 1)P (αβ)n (u) = (n+ α+ β + 1)P (α+1β)





n (u2)P(αβ)n (u1)

u1 minus u2

=n+ α+ β + 12n+ α+ β + 1

P (α+1β)n (u1)P (α+1β)

n (u2) +n+ β

2n+ α+ β + 1

times(1minus u1)P

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2

(29)


α+ β + 12



n (u)

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1144 A I Bufetov




(α+1β)n (u2)P


u1 minus u2

= minus n

n+ α+ 1P (α+1β)

n (u1)P (α+1β)n (u2) +

2n+ α+ β + 12(n+ α+ 1)


timesP

(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2 (30)




n (u2)P(αβ)n (u1)

u1 minus u2

=(α+ 1)(2n+ α+ β + 1)

(n+ α+ 1)(2n+ α+ β + 1)P (α+1β)

n (u1)P (α+1β)n (u2) +

n+ β

2(n+ α+ 1)


(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2


(2n+ α+ β + 2)P (α+1β)n (u) = (n+ α+ β + 2)P (α+2β)



P(α+1β)n (u1)P


(α+1β)n (u2)P


u1 minus u2=

n+ α+ β + 22n+ α+ β + 2

timesP

(α+2β)n (u1)P


(α+2β)n (u2)P


u1 minus u2 (32)



L(αβn)Jac = span













nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography







nsuml=0

(2l + α+ β + 1)2α+β+1

Γ(l + 1)Γ(l + α+ β + 1)Γ(l + α+ 1)Γ(l + β + 1)

P(α)l (u)Q(α)

l (ν)

=12


(ν minus u)+

2minusαminusβ

2n+ α+ β + 2




n (u)ν minus u








11minus u

+Γ(α)Γ(n+ 2)Γ(n+ α+ 1)


Jac



(s0)n



n


k(s)n =

Γ(2n+ s+ 1)2nn Γ(n+ s+ 1)

h(s)n =

int 1

minus1


2s+1

2n+ s+ 1


K(s)n (u1 u2) =

nminus1suml=1

P(s)l (u1)P

(s)l (u2)

h(s)l


1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1146 A I Bufetov


n

K(s)n (u1 u2)

=n(n+ s)


(s)n (u1)P


(s)n (u2)P

(s)nminus1(u1)

u1 minus u2

(35)



radicy1)Js(

radicy2)minus

radicy2Js+1(

radicy2)Js(

radicy1)

2(y1 minus y2)


Js(y1 y2) =14

int 1

0

Js

(radicty1

)Js

(radicty2

)dt (36)










δx





B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography




B Conf(E) rarr R





ϕ(X) =sumxisinX

ϕ(x)





ϕ1 ϕ2 ϕl





1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1148 A I Bufetov





χBKχBI1






l = det(

1 +lsum

j=1


i Bi

)





KF =

radicdFlowastmicro

dmicroFlowastK

radicdFlowastmicro

dmicro



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography



KF (x y) =

radicdFlowastmicro


radicdFlowastmicro

dmicro(y)



FlowastPK = PKF


ϕ Fminus1(x)

radicdFlowastmicro

dmicro(x)

sub L2(Emicro)


Ψg(X) =prodxisinX

g(x)








1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1150 A I Bufetov




Ψg dPK = det(1 +

radicg minus 1K

radicg minus 1

)gt 0


PB(gK) =ΨgPKint

Conf(E)Ψg dPK


radicg minus 1K

radicg minus 1 is



radicg L







andPQ|Conf(EEprime)





Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography




Proposition B6 If












Ψg dPΠ = det(1 +

radicg minus 1 Π

radicg minus 1

)


= PΠg




1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1152 A I Bufetov





formulamicro(s)


where


l=m0

Γ(l + s)Γ(n+ l + s)


πmnm1n1











i=1



nsumi=1

|ξi|2 u2i

1 + u2i




i=1

|ξi|2 u2i

1 + u2i

= 1 +nsum

i=1

|ξi|2

1 + u2i


nsumi=1

|ξi|2

1 + u2i


dξ =mprod

i=1

(1+u2i )

πn

Γ(n)

int +infin

0


where


i=1

|ξi|2

1 + u2i

(39)



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography



0



We introduce a map



mn (z) = (πm+1nmn (z) r(m+1n)(z))






mn

)lowastmicro

(s)m+1n = micro(s)

mn times P (m+1ns)


πmn+1mn (z)

)t = πm+1nmn (zt)




mn (z) =(πmn+1

mn (z) r(mn+1)(z))


mn

)lowastmicro

(s)mn+1 = micro(s)

mn times P (n+1ms)



by the formula



1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography

1154 A I Bufetov




infinprodl=0


) (40)

Bibliography





































1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography






















1156 A I Bufetov











Alexander I Bufetov











sect1 Introduction








144 Scaling limits










1111 Approximation






























Bibliography