1

VICIOUS WALKand

RANDOM MATRICES

Makoto KATORI(Chuo University, Tokyo, Japan)

Joint Work withHideki Tanemura (Chiba University) and

Taro Nagao (Nagoya University)

Nonequilibrium Statistical Physics of Complex Systems Satellite Meeting of STATPHYS 22 in Seoul, Korea, KIAS International Conference Room, 29 June-2 July 2004

21. INTRODUCTION

• Consider a Standard Brownian Motion B(t) in One Dimension. Stochastic Properties of the Variation .))((,0)( 2 dttdBtdB

•Transition Probability Density from x at time s to yy at time t (>s) is given by

.)(2

)(exp

)(2

1),;,(

2

st

yx

stytxsp

•This solves the Heat Equation (Diffusion Equation);

).(),;,(lim),,;,(2

1),;,( 2

2

yxytxspytxspx

ytxspt st

B(s)=x

B(t)= ?

3•Next we consider a pair of independent Brownian Motions, )(~),( tBtB

.00)(~)()(~)( ttBdtdBtBdtdB

We define a complex-conjugate pair of complex-valued stochastic variables,

)(~

1)(2

1)()(

~1)(

2

1)( * tBtBtxtBtBtx

•By definition

)(~

1)(2

1)()(

~1)(

2

1)( * tBdtdBtdxtBdtdBtdx

which give

dttBdtdB

tBdtdBtBdtdBtdxtdx

tBdtdBtBdtdBtdx

tBdtdBtBdtdBtdx

22

222

2

222

2

)(~)( 2

1

)(~1)(2

1)(~1)(

2

1)()(

0)(~)( 2

1)(~1)(

2

1)(

0)(~)( 2

1)(~1)(

2

1)(

*

We have a correlation between the complex-conjugate pairs.

42. HERMITIAN MATRIX-VALUED PROCESS AND DYSON’S BROWNIAN MOTION MODEL

•Let be mutually independent (standard one-dim.) Brownian motions started from the origin. Define

,,1),(~

),( NjitBtB ijij

)()(~

2

1)(0

)()(~

2

1

)(

)()(2

1)()(

)()(2

1

)(

jitB

ji

jitB

ta

jitB

jitB

jitB

ts

ij

ij

ij

ij

ii

ij

ij

•Consider the Hermitian Matrix-Valued stochastic processNN

NjiijijNjiij tatstt

,1,1

)(1)()()( That is,

)(...)(1)()(1)()(1)(

...............

...............

)(1)(...)()(1)()(1)(

)(1)(...)(1)()()(1)(

)(1)(...)(1)()(1)()(

)(

332211

333323231313

222323221212

111313121211

tstatstatstats

tatststatstats

tatstatststats

tatstatstatsts

t

NNNNNNNN

NN

NN

NN

5

•Consider the variation of the matrix, It is clear that

And by the previous observation, we find that

They are summarized as

• Since is a Hermitian matrix-valued process, at each time t there is a Unitary Matrix , such that

where the eigenvalues are in the increasing order

• We can regard

as an N-particle stochastic process in one dimension.

)(t

Njiij tdtd

,1

)()(

),1(0)( Njitd ij

)1()(

)1()()()()(

)1(0)(

2

*

2

Nidttd

Njidttdtdtdtd

Njitd

ii

ijijjiij

ij

),,,1()()( Nnkjidttdtd jkinknij

Njiij tutU ,1))(()(

)(),....,(),(diag)()()()( 21 tttttUttU N

,0),(....)()( 21 tttt N

N

N tttt Rλ )(),....,(),()( 21

6

QUESTIONBy the diagonalization of the matrix, what kind of interactions emerge

among the N particles in the process ?)(tλ

•From now on we assume that

•And we consider the following conditional configuration-spaceof one-dim. N particles,

(This is called the Weyl chamber of type . )

)0(....)0()0( 21 N

N

NA

N xxx ....: 21RxW

1NA

7ANSWER 1 (by Dyson 1962)1. For all t > 0, with Probability 1.2. The process is given as a solution of the stochastic differential equations,

where are independent standard one-dim. Brownian motions .

A

Nt Wλ )(

,0,1)()(

1)()(

,1:tNidt

tttdBtd

ijNjj jiii

)(tBi )1( Ni

• This process is called Dyson’s Brownian motion model.

• Strong repulsive forces emerge among any pair of particles

distance particle

1

8

• Let

•Consider

It solves the Fokker-Planck (FP) equation in the form

)(),....,(),()( , )1( )h( ln1

)(

)sdifference ofproduct ( )()h(

21,1:

1

xxxxbxx

x

N

iijNjj

ji

i

iNji

j

bbbNixxx

b

xx

. )...., , ,( ), ...., , ,( where

, )( to)( fromdensity y probabilit transition),;,p(

2121 NN yyyxxx

tsts

yx

yλxλyx

),;,p()(),;,p(2

1),;,p( yxxbyxyx xx tststs

t

ANSWER 2Introduce a determinant

Then the solution of the FP equation is given by

• If (all particles starting from the origin)

tj

yi

x

ijijNji txytxytt

2/2)(

,1e

2

1)|,G( with )|,G(det)|,f(

xy

. , ,0for )h()|,f()h(

1),;,p( A

Ntsstts Wyxyxyx

yx

,0at )0,....,0,0( s0x

N

i

NN

ii

N

iCytC

tt

1

2/1

1

2222

1

2/2

. )()2( , || where)h(2

||exp),;,0p( yy

yy0

9REMARK 1: When all the particles are starting from the origin 0,

Dyson’s Brownian motion model = NONCOLLIDING Diffusion Particle Systems

2)(e 2

1),;,0p(

11

2/2

iNji

j

N

k

tk

yyy

tt

y0

Product of Independent Gaussian Distributions

Strong Repulsive Interactions

. 0),;,0p( ,0|| As y0 tyy ij

10REMARK 2: Here we set N = 3 as an example.

Dyson’s Brownian motion model = a Free Fermion System

)|,G()|,G()|,G(

)|,G()|,G()|,G(

)|,G()|,G()|,G(

det)(

)(

,,,;,,,p

333231

232221

131211

321321

xystxystxyst

xystxystxyst

xystxystxyst

xx

yy

yyytxxxs

jiij

jiij

h-transform in the sense of Doob (Probab.Theory)

a stochasic version of Slater determinant(Karlin-McGregor formula in Probab.Theory)(Lindstrom-Gessel-Viennot formula in Combinatorics )

113. VICIOUS WALKSAs Temporally Inhomogeneous Noncolliding Particle Systems

Physical Motivations to Study Vicious Walker Models

•As a model of Wetting or Melting Transitions(Fisher (JSP 1984))

•As a model of Commensurate-Incommensurate Transitions(Huse and Fisher (PRB 1984))

12

•As a model of Directed Polymer Networks(de Gennes (J.Chem.Phys. 1968), Essam and Guttmann (PRE 1995))

(a) polymer with star topology (b) polymer with watermelon topology

From the viewpoint of solid-state physics,we want to treat Large but Finite system with Boundary Effects.

13

t

N

dttAN

time toupother each with collide

from starting particlesBrownian Prob

)|,f( ),N(

not dox

yxyxw

NONCOLLIDING PROBABILITY:We can see that

Noncolliding Condition Imposed for Finite Time-Period (0,T]•We introduce a parameter T, which gives the time period in which the noncolliding condition is imposed.•The transition probability density of the Noncolliding Brownian Motions during time T from the state x at time s to the state y at time t is

•The following are satisfied.

A

N ,T, tssT

tTstts Wyx

x

yxyyx

0for ),N(

),Ν()|,f(),;,q(

. , ,0 , ),;,q(),;,q(),;,q( )3(

, 1),;,q( )2( , )(),;,q( lim )1(

A

N

A

Nst

AN

AN

Tutsusdutts

dtsts

Wzxzxyzyyx

Wxyyxyxyx

W

W

14

A

N ,T, tssT

tTstts Wyx

x

yxyyx

0for ),N(

),N()|,f(),;,q(

15Estimations of Asymptotics

By using the knowledge of symmetric functions called Schur Functions,we can prove the following three facts.[1] Exponent of Power-Law Decay of the Noncolliding Probability: For fixed initial positions

[2] The limit gives the Temporally Homogeneous process.

A

NWx

. in )1(4

1 with )h(~),N( tNNtt xx

T

. 0for ),;,p(),;,q(lim

tststsT

yxyx

. )1(

1

)polynomial(Schur detdet)(with

, )()()h()h(

expdet

1

,1,1

)( :

,1

N

ii

jN

iNji

jN

iNji

N

jiNji

iNa

xxs

ssayx

j

x

yxyx

SCHUR FUNCTION EXPANSION

16

[3] The limit of the transition probability density is well defined as follows.0x

N

j

NNNN

jCtTtC

tT

tt

1

2/

2

2

2

2/24/)1(

0||

. )2/(2 with ),N()h(2

||exp

),;,0q( lim),;,0q(

yxy

yxy0x

Two Limit Cases(1) Case t=T Since

we have

(2) Case

, 1)()|,0f(),0N( AN

AN

ddWW

yyzyyzy

. )h(2

||exp),;,0q(

2

2

4/)1(

yy

y0

TC

TT

NN

T

2)h(2

||exp) ; ; ,0 p() , ; ,0 q( lim

2

1

2/2

yy

y0y0

tC

ttt

N

T

17Case t = T

Case T

18

Transition in Time of Particle Distribution• This observation implies that there occurs a transition.

For a finite but large T

Ttxxyt

yt jNkj

k

N

ii

0for )(

2

1exp),;,0q( 2

11

20

As the time t goes on from 0 to T

TtxxyT

T jNkj

k

N

ii

at )(

2

1exp),;,0q(

11

2y0

19

Three Standard (Wigner-Dyson) Random Matrix Ensembles

[1] The distribution of Eigenvalues of Hermitian Matrices in the Gaussian Unitary Ensemble (GUE) is given in the form

[2] The distribution of Eigenvalues of Real Symmetric Matrices in the Gaussian Orthogonal Ensemble (GOE) is given in the form

[3] The distribution of Eigenvalues of Quternion Self-Dual Hermitian Matrices in the Gaussian Symplectic Ensemble (GSE) is given in the form

2)(2

1exp

11

22 j

Nkjk

N

ii

NN

NN

NN

)(2

1exp

11

22 j

Nkjk

N

ii

4)(2

1exp

11

22 j

Nkjk

N

ii

204. PATTERNS of NONCOLLIDING PATHS AND RANDOM MATRIX THEORIES

4.1 STAR CONFIGURATIONS• There occurs a transition in distribution from GUE to GOE.

• This temporal transition can be decribed by the Two-Matrix Model of Pandey and Mehta, in which a Hermitian random matrix is coupled with a real symmetric random matrix. See Katori and Tanemura, PRE 66 (2002) 011105/1-12.• Techniques developed for multi-matrix models can be used to evaluate the dynamical correlation functions. Quaternion determinantal expressions are derived. See Nagao, Katori and Tanemura, Phys. Lett. A307 (2003) 29-35.• Using the exact correlation functions, we can discuss the scaling limits ofinfinite particles and the infinite time-period . See Katori, Nagao and Tanemura, Adv.Stud.Pure Math. 39 (2004) 283-306.

N T

21

4.2 Watermelon Configurations• Consider a finite time-period [0,T] and set y=0 at the initial time t=0 and the final time t=T.• The transition probability density is given as

• The distribution is kept in the form of GUE.• Only the variance changes as a function of t as .

2)h(

/12

||exp1

1),;,0(q

2

1

2/2

watermelon yy0

Ttt

y

T

tt

Ct

N

T

tt 12

224.3 Banana Configurations

• Consider 2N particle system. Set y=0 at the initial time t=0. At the final time t=T, we assume the following Pairing of Particle Positions.

• The transition probability density is given by

• As , there occurs a transition from the GUE distribution to the GSE distribution.

. .... with , .... , , 12312124321 NNN yyyyyyyyy

. )|,G()|,G(det),(N where

, , ,0for ),(N

),(N)|,f(),;,0(q

1,21

2

banana

banana

bananabanana

iji

ijNjNi

A

N

xytt

xxytxt

TtsT

tTtt

AN

W

Wyxx

yxyyx

Tt 0

234.4 Star Configurations with Absorbing Wall

• Put an Absorbing Wall at the origin. Consider the N Brownian particles started from 0 conditioned never to collide with each other nor to collide with the wall. • This is identified with the h-transform of the N-dim. Absorbing Brownian motion in

• For , we can obtain a process showing a transition from the class C distribution of Altland and Zirnbauer (1996);

to the class CI distribution (studied for a theory of quantum dots)

). typeofchamber (Weyl ....0: 21 NN

NC

N Cxxxx RW

T

Ttyyyt

tN

kki

Njij

C

0for )(

2

||exp),;,0(q

1

222

1

22y

yx

.at )(2

||exp),;,0(q

1

2

1

22

TtyyyT

TN

kki

Njij

C

yyx

244.5 Star Configurations with Reflection Wall

• Put a reflection wall at the origin. Consider the N Brownian particles started form 0 conditioned never to collide with each other.• This is identified with the h-transform of the N-dim. Absorbing Brownian motion in

). typeofchamber (Weyl ....|:| 21 NN

ND

N Dxxxx RW

For , we can obtain a process showing a transition from the class D distribution of Altland and Zirnbauer (1996);

to the ``real” class D distribution

T

Ttyyt

t iNji

jD

0for )(

2

||exp),;,0(q 22

1

22y

yx

.at )(2

||exp),;,0(q 2

1

22

TtyyT

T iNji

jD

yyx

254.6 Banana Configurations with Reflection Wall

• Put a reflection wall at the origin. • Consider the 2N Brownian particles started from 0 in Banana configurations.

• For , we can obtain a process showing a transition from the class D distribution of Altland and Zirnbauer

To the class DIII distribution.

Ttyyt

t iNji

jD

0for )(

2

||exp),;,0(q 22

1

22

, banana yyx

T

.at )(||

exp),;,0(q1

12

2

121

2

12

24

oddbanana , Ttyyy

TT

N

kki

Njij

D

yyx

265. CONCLUDING REMARKS• There are 10 CLASSES of Gaussian Random Matrix Theories.

Standard (Wigner-Dyson) GUE Star configurations GOE GSE Banana configurationsNonstandard (chiral random matrices) Particle Physics of QCD chGUE chGOE Realized by Noncolliding Systems of chGSE 2D Bessel processes and Generalized MeandersNonstandard (Altland-Zirnbauer) Mesoscopic Physics with Superconductivity class C class CI Star config. with Absorbing Wall class D class DIII Banana config. With Reflection Wall

All of the 10 eigenvalue-distributions can be realized by the Noncolliding Diffusion Particle Systems (Vicious Walks).

See Katori and Tanemura, math-ph/0402061, to appear in J.Math.Phys.(2004)

27

REMARK 3.Relations between Random Matrices and Vicious Walks are very useful to analizeother nonequilibrium models, e.g. Polynuclear Growth Models. (See Sasamoto and Imamura, J. Stat. Phys. 115 (2004))

Future Problems• Calculate the dynamical correlation functions and determine the scaling limits of all these (temporally inhomogeneous) noncolliding systems. (some of them are done by Forrester, Nagao and Honner (Nucl.Phys.B553 (1999) )• The 10 classes are related with the diffusion processes on the flat symmetric spaces. Extensions to the noncolliding systems of diffusion particles in the space with positive curvature Ref: Circular Ensembles of random matricesand in the space with negative curvature. Ref. Theory of Quantum Wire: DMPK equations Beenakker and Rejaei, PRB 49 (1994), Caselle, PRL 74 (1995) Ref. Symmetric Spaces: Caselle and Magnea, Phys.Rep. 394 (2004).

TN and

28

[1] M.Katori and H. Tanemura, Scaling limit of vicious walkers and two-matrix model, Phys.Rev. E66 (2002) 011105.[2] M.Katori and H. Tanemura, Functional central limit theorems for vicious walkers, Stoch.Stoch.Rep. 75 (2003) 369-390;arXiv.math.PR/0204386.[3] T.Nagao, M.Katori and H.Tanemura, Dynamical correlations among vicious random walkers, Phys.Lett.A307 (2003) 29-35.[4] J.Cardy and M.Katori, Families of vicious walkers, J.Phys.A Math.Gen. 36 (2003) 609-629.[5] M.Katori, T. Nagao and H. Tanemura, Infinite systems of non-colliding Brownian particles, Adv.Stud.in Pure Math. 39 ``Stochastic Analysis on Large Scale Interacting Systems”, pp.283-306, Mathematical Society of Japan, 2004; arXiv.math.PR/0301143.[6] M.Katori and N. Komatsuda, Moments of vicious walkers and Mobius graph expansion, Phys.Rev. E67 (2003) 051110.[7] M.Katori and H. Tanemura, Noncolliding Brownian motions and Harish-Chandra formula, Elect.Comm.in Probab. 8 (2003) 112-121; arXiv.math.PR/0306386.[8] M.Katori, H.Tanemura, T.Nagao and N.Komatsuda, Vicious walk with a wall, noncolliding meanders and chiral and Bogoliubov-deGennes random matrices, Phys.Rev. E68 (2003) 021112.[9] M.Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, to appear in J.Math.Phys.; arXiv:math-ph/0402061.

References