HOLE PROBABILITY AND LARGE DEVIATIONS IN THEDISTRIBUTION OF THE ZEROS OF GAUSSIAN RANDOM

HOLOMORPHIC FUNCTIONS.

by

Scott E. Zrebiec

A dissertation submitted to the Johns Hopkins University in conformity with therequirements for the degree of Doctor of Philosophy.

Baltimore, MD

March, 2007

Contents

1 Introduction 1

2 Common results 8

2.1 Defining a Gaussian random holomorphic function . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The radius of convergence of a random power series. . . . . . . . . . . . . . . . . . . . . . 15

2.3 Invariance laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Zero sets of m variable holomorphic functions and currents . . . . . . . . . . . . . . . . 19

2.5 The expected zero set of a Gaussian random holomorphic function. . . . . . . . . . 23

3 New results concerning large deviations in the distribution of zeros of

a Gaussian entire function 27

3.1 An estimate for the growth rate of Gaussian entire functions . . . . . . . . . . . . . . . 27

3.2 Large deviations in the value of the Nevanlinna characteristic function of a

Gaussian entire function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Large deviations in the volume of the zero sets of Gaussian entire functions, and a

derivation of the order of the decay of the hole probability. . . . . . . . . . . . . . . . . . 41

4 New results concerning large deviations of the distribution of zeros of

a Gaussian random SU(m+1) polynomial 46

4.1Large deviations of the maximum value of a random SU(m+1) polynomial on a

ball of radius r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv

4.2 Large deviations in the value of the Nevanlinna characteristic function of a

Gaussian SU(m+1) polynomial of degree N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Main results concerning random SU(m+1) polynomials of degree N . . . . . . . . . 56

Bibliography 60

v

ACKNOWLEDGEMENT

There are many people I would like to thank for helping me get through graduate

school. First and foremost, I would like to thank Bernie Shiffman for many useful

discussions and for being a mentor and role model. I would also like to thank Catherine

Coker for supporting me during this lengthy and stressful process, and for putting up with

me when I would rush home to correct a typo. Additionally, many faculty and graduate

students have helped me with various parts of my work and or life, especially Brian

Macdonald, Bruce Johnson, Chika Mese, Giuseppe Tinaglia, Jian Song, John Baber,

Mike Krebs, Steve Zelditch and many others.

DEDICATION

I would like to dedicate this work to my late grandfather Francis Kelly who made

the world and my world a better place.

iii

1. Introduction

Random polynomials and random holomorphic functions are studied as

a way to gain insight into difficult problems in physics, geometry and ana-

lytic number theory, and because of these applications, random holomorphic

functions have been studied in a flurry of recent research. Gaussian random

holomorphic functions are defined as

ψα(z) =∑

αjψj(z), (1)

where αj are independent identically distributed standard centered com-

plex Gaussian random variables and for all j, ψj ∈ O(Mm), M an m man-

ifold. A particularly interesting case of random holomorphic functions is

when the random functions can be defined so that they obey certain in-

variance laws with respect to the natural isometries of the manifold Mm.

The following choices for ψj are particularly well studied for this reason,

([BSZ2], [ST1]):

Name Related Isometries Random Expected

space for (m = 1) function zero set

entire flat Cm z 7→ eiθz + ζ∑

j∈Nm

αjzj

√j!

i2π

∑dzi ∧ dzi

A ∈ U(m)

SU(m + 1) CPm z 7→ az+b−bz+a

,N∑

|j|=0

αj

(N

j

) 12

zj Ni2π

∑dzi∧dzi

(1+|z|2)2

polynomials |a|2 + |b|2 = 1

SU(1, 1), D ⊂ C1 z 7→ az+bbz+a

,∑

αjzk 1

2πidz∧dz

(1−|z|2)2

analytic on D |a|2 − |b|2 = 1 z ∈ C1

random (Mm, ω) None∑

j αjSj(z) Asymptotic

holomorphic L > 0 in Sj any o.n. basis to

sections M cmpt Kahler general of H0(M,LN ) N · ω

While the topic of random holomorphic functions, and the zeros of said

functions, is of recent interest there is also an old history complete with

many results from the first half of the twentieth century, which is now ex-

periencing a renaissance. Historically, many mathematicians have studied

various properties of random functions including Polya, Erdos-Renyi, Salem-

Zygmund, Paley-Wiener. Additionally, the study of the zeros of random

holomorphic functions was born with the work of Kac and Rice, who deter-

mined a formula for the expected distribution of zeros of real polynomials

in a certain cases, ([Kac], [Ric]), work which has been subsequently gener-

alized throughout the years, [EK]. An excellent reference for other results

regarding the general properties of random functions is Kahane’s text, [Kah].

More recently there was a series of papers of Offord which is particularly

relevant to questions involving the hole probability of random holomorphic

2

functions and the distribution of values of random holomorphic functions,

[Off1], [Off2].

These classical results have been built upon with recent research which

statistically studies the behavior of zeros of random holomorphic functions.

In particular first and second moments for the number of zeros of a random

holomorphic function have been computed, as well as a central limit law

([ST1], [SZ]). Additionally, recent work suggests that the zero sets of ran-

dom polynomials and holomorphic functions are much more natural objects

than they may initially appear. For example, Bleher, Shiffman and Zelditch

have shown that for any positive line bundle over a compact complex man-

ifold, the random holomorphic sections to LN (defined intrinsically) have

universal high N correlation functions, [BSZ2].

In addition to a plethora of results describing the typical behavior, there

have also been several results in 1 (real or complex) dimension for Gauss-

ian random holomorphic functions where there is a large deviation in the

number of zeros present, including the event where there is a hole. A hole

is an event where there are no (complex) zeros present in a domain where

many are expected. Let Holer be the event: f , in a class of holomorphic

functions, such that ∀z ∈ B(0, r), f(z) 6= 0, where B(0, r) = z : |z| < r.

For the complex zeros in one complex dimension, there is a general upper

bound for the hole probability of a Gaussian random holomorphic func-

tion: Prob(Holer) ≤ e−cρ(B(0,r)), ρ(Ω) =∫

ΩE[Zψα ] as in theorem 2.12,

[Sod]. In one case this estimate was shown by Peres and Virag to be sharp:

Prob(Holer) = e−ρ(B(0,r))

24+o(ρ(B(0,r))), [PV]. These last two results on hole

3

probability might suggest that when the expected zero set of a Gaussian

random holomorphic functions is invariant with respect to the local isome-

tries then the rate of decay of the hole probability would be the same as

that which would be arrived at if the zeros where distributed according to

an appropriate Poisson process. However, as the zeros repel in 1 dimension

[BSZ1], one might expect there to be a quicker decay for hole probability

of a random holomorphic function. This is the case for Gaussian entire flat

functions in 1 complex variable, [ST2] :

Prob(Holer) ≤ e−c1r4

= e−cρ(B(0,r))2 , and Prob(Holer) ≥ e−c2r4

= e−cρ(B(0,r))2

Additionally questions related to hole probability have been answered for

a specific class of real Gaussian polynomials, [DPSZ], [LS]. However, there

had never been any non trivial results concerning the order of the decay of

the hole probability in more than one complex variable.

In this thesis work we will examine just such problems by studying large

deviations of the zero set of Gaussian entire flat functions, in n variables,

and random SU(m+1) polynomials. This will be done in three sections. In

the first section we shall state, and where necessary prove results which are

typical of those found in the literature, even though in certain cases these

typical results will need to be adopted to our m variable cases. We will also

discuss various technical details concerning Gaussian random holomorphic

functions. The second section will concern a new and interesting result

concerning the decay of the hole probability of Gaussian entire flat functions.

In the third section we will, using several lemmas from section 2, derive the

4

order of the decay of the hole probability for Gaussian random SU(m + 1)

polynomials. This last result is also a new and interesting result for the one

variable case, or when m = 1, as well as for the higher dimensional case.

More specifically in section 2, we will prove two main results concerning

Gaussian entire flat functions and the volume of the zero sets of these func-

tions. To describe the volume of the zero set we will be using the unitegrated

counting function which is defined as:

nψα,N (r) =1

(m− 1)!r2m−2πm−1· Volume of the zero set of ψα,N

⋂B(0, r),

where B(0, r) = z ∈ Cm : |z| < r.

The first two of these main results generalize Sodin and Tsirelison’s re-

sult for Gaussian entire flat functions in one variable, [ST2]. The first of

these main results is:

Theorem 1.1. If ψα(z) is a Gaussian entire flat function, i.e.

ψα(z1, z2 . . . , zm) =∑j∈Nm

αjzj11 z

j22 . . . zjmm√j1! · jm!

,

where αj are independent identically distributed centered complex Gaussian

random variables, then for all δ > 0, there exists cδ,m > 0 and Rm,δ such

that for all r > Rm,δ

Prob

(∣∣∣∣nψα(r)− 1

2r2

∣∣∣∣ ≥ δr2

)≤ e−cδr

2m+2

where nψα(r) is the unintegrated counting function for ψα.5

We will often use the standard multi index notation to simplify our

formulas. Specifically, if z ∈ Cm and j ∈ Nm then

zj = zj11 zj22 · . . . · zjmm

j! = j1! · j2! · . . . · jm!(N

j

)=

N !

j! · (N − |j|)!

As an example, a Gaussian entire flat function may be written as ψα(z) =∑αj

zj√j!.

Theorem 1.2. If ψα(z) is a Gaussian entire flat function and if

Holer = ∀z ∈ B(0, r), ψα(z) 6= 0,

then there exists Rm, cm, and Cm > 0 such that for all r > Rm

e−Cmr2m+2 ≤ Prob(Holer) ≤ e−cmr2m+2

The proof of theorems 1.1 and 1.2, will use value distribution theory,

which relates the expected number of zeroes with the rate of growth of the

maximum of a function, basic probability theory, and standard techniques

of complex analysis. Additionally, an invariance rule for Gaussian entire

flat functions with respect to translation will be used. These results, using

the mainly same techniques, were already proven in the case where m=1

by Sodin and Tsirelson, [ST2]. Additionally, the results of section 2 will be

appearing in the Michigan Mathematical Journal, [Zre1].6

In section 3 of this thesis we will consider the class of random polynomials

whose zeros are distributed on CPm according to the Fubini-Study measure.

The random functions which will be studied here are called Gaussian random

SU(m+ 1) polynomials and can be written as:

ψα,N(z) =N∑

|j|=0

αj

√(N

j

)zj (1)

where ∀j, αj, are independent identically distributed standard centered

complex Gaussian random variables (mean 0 and variance 1). For these

Gaussian random SU(m+ 1) polynomials we will prove two results regard-

ing the distribution of the zeros, by using similar techniques to those used

to solve the m variable entire flat case:

Theorem 1.3. If ψα,N is a degree N Gaussian random SU(m+ 1) polyno-

mial,

ψα,N(z) =N∑

|j|=0

αj

√(N

j

)zj,

where αj are independent identically distributed centered complex Gaussian

random variables, and if nψα,N(r) is the unintegrated counting function then

for all ∆ > 0, and r > 0 there exists A∆,r,m and N∆,r,m such that for all

N > N∆,r,m

Prob

(∣∣∣∣nψα,N(r)− Nr2

1 + r2

∣∣∣∣ ≥ ∆N

)< e−A∆,r,mN

m+1

.

Typically, nψα,N(r) should be about Nr2

1+r2. Theorem 1.3 gives an upper

bound on the rate of decay of the hole probability, and we will be able to

prove a lower bound for the decay rate of the same order:7

Theorem 1.4. Let ψα,N be as in theorem 1.3, and let

HoleN,r = α : ∀z ∈ B(0, r), ψα,N(z) 6= 0,

then there exists c1,r,m, c2,r,m > 0 and Nr such that for all N > Nr,m

e−c2,rNm+1 ≤ Prob(HoleN,r) ≤ e−c1,rNm+1

As an immediate consequence of this result, the order of the probability

specified in the Theorem 1.3 is the correct order of decay.

Random SU(m + 1) polynomials of the form studied here are the sim-

plest examples of a class of natural random holomorphic sections of large

N powers of a positive line bundle on a compact Kahler manifold. Most of

the results stated in this paper may be restated in terms of Szego kernels,

whose leading order scaling asymptotics exhibit universal behavior in the

large N limit, when appropriately scaled. Hopefully, this paper may provide

insight into proving a similar decay rate for this more general setting. This

has already been done for other properties of random holomorphic sections,

e.g. correlation functions, [BSZ2].

2. Common results

2.1. Defining Gaussian random holomorphic functions. There are

two general, and equivalent perspectives of how to define a Gaussian ran-

dom holomorphic function: either one establishes a Gaussian probability

distribution on a Hilbert space of functions or alternatively, or one uses line

(1) of the introduction. These two perspectives are equivalent, but we shall8

always explicitly use the second, while occasionally implicitly using the first.

By doing this we may avoid the technical, abstract definitions of Gaussian

Hilbert spaces and Gaussian fields.

Our definition of a Gaussian random holomorphic function makes use of

the standard definition of Gaussian random variables:

Definition 2.1. A Gaussian random variable is a complex-valued random

variable with the following probability distribution function:

dν(z) =1

πσe−|z|2

σ2 dm(z),

This is a stronger definition than most in that it assumes that the mean

is zero, and is often called a centered complex Gaussian random variable. In

this paper we will only use (centered complex) Gaussian random variables,

and thus we will not bother including the two words centered and complex

when discussing Gaussian random variables. Additionally, we will call a

Gaussian random variable whose variance is 1, a standard Gaussian random

variable. As is always the case, a (complex) Gaussian random variable

can be written as the sum of two independent jointly normal variables for

R2, where the variance of the real part is the same as the variance of the

imaginary part (these random variables are independent).

A straightforward computation shows that Gaussian random variables

have the following moments:9

E[z] :=∫

C zdν(z) = 0

E[1] :=∫

C dν(z) = 1

E[|z|2] := σ2 = V ar(z)

∀p > 0, E[|z|p] = σ(Γ(p+12

))1p

To finish the definition of a Gaussian random holomorphic function we

will now choose ψj to be a specific orthonormal basis of one of two Hilbert

spaces of holomorphic functions. The first of these two Hilbert spaces relates

to Gaussian entire flat functions and is:

H2(Cn, e−|z|2

dm(z)) =

f ∈ O(Cm) :

∫z∈Cm

|f(z)|2e−|z|2dm(z) <∞,

where dm is Lebesque measure. This Hilbert space has the following two or-

thonormal bases:z

j11 ·zj2

2 ·...zjnn√

j1!·...·jm!

j∈Nm

= ψj(z)Nm ande−

12|ζ|2−zζ (z+ζ)j

√j!

j∈Nm

.

We will now prove that the previously mentioned collections of functions

are in fact orthonormal bases for H2(Cm, e−|z|2dm(z)):

Proof. It suffices to show that for all ζ ∈ Cm the second collection of func-

tionse−

|ζ|22−zζ (z+ζ)j

√j!

j∈Nm

is an orthonormal basis:∥∥∥∥e− 12|ζ|2−zζ (z + ζ)j√

j!

∥∥∥∥ =1

(j!πm)

∫Cm

|z + ζ|2je−|ζ|2−zζ−ζze−|z|2dm(z)

=1

(j!πm)

∫Cm

|z|2je−|z|2dm(z)

=1

(j!)

∫R+,m

uje−u, by setting u = r2

= 1e−u|u=∞u=0 = 1, by integrating by parts

By symmetry this basis, is orthogonal, and as any holomorphic function

can be written as a (convergent) power series, this is a basis.10

Our second set of basis functions comes from a Hilbert space of poly-

nomials and is the set of polynomials in one variable whose degree is less

than or equal to N , PolyN along with the SU(m+1) invariant norm, [BSZ2]:

‖f‖2N :=

N +m!

N !πm

∫z∈Cm

|f(z)|2 dm(z)

(1 + |z|2)N+m+1,

where dm is just the usual Lebesque measure. For this norm√(

Nj

)zj

is an orthonormal basis, as is

√(Nj

) m∏k=0

(m∑l=1

ak,lzl + ak,0)jk

, where A =

(ak,l) and A ·AT = I, and j0 = N −|j|. Specifically, one alternate orthonor-

mal basis is, for any ζ ∈ C1,

√(Nj

)(z1−ζ√1+|ζ|2

)j1 (1+ζz1√1+|ζ|2

)N−|j|∏mk=2 z

jkk

|j|≤N

.

Clearly, by line (1), a Gaussian random SU(m+1) polynomial is defined

as, ψα,N(z) =

|j|=N∑|j|=0

αjψj(z), where αj are i.i.d. standard complex Gaussian

random variables, and ψj is the first orthonormal basis. Any basis for

PolyN could have been used and the Gaussian random SU(m + 1) polyno-

mials would be probabilistically identical, as for αj a sequence of i.i.d.

Gaussian random variables there exists another sequence of i.i.d. Gaussian

random variables, α′j, such that

∑αj

√(N

j

)zj =

N∑|j|=0

α′j

√(N

j

)(z1 − ζ√1 + |ζ|2

)j1 (1 + ζz1√1 + |ζ|2

)N−|j| N∏k=2

zjkk .

11

We will now prove that these collections of functions are in fact orthonor-

mal bases for PolyN :

Proof. By making a specific SU(m+1) transformation we may get from the

first basis to the second:⟨√(N

j

)zj,

√(N

k

)zk

⟩=

(N +m)!

N !πm

∫Cm

(N

j

)zj11 z

k11 z

j zk

(1 + |z1|2 + |z|2)N+2dm(z),

where z = (z2, z3, . . . , zm) and j = (j2, j3, . . . , jm). We now make the fol-

lowing SU(m+ 1) transformation:

Let a =1√

1 + |ζ|2and b =

−ζ√1 + |ζ|2

, which guarantees that |a|2+ |b|2 = 1.

Let z1 = aw1+b−bw1+a

.

Let zk = wk

−bw1+a, for all k 6= 1.

Hence, dz1 = dw1

(−bw1+a)2,

and dzk = dwk

(−bw1+a)+ −bwk

(−bw1+a)2dw1.

Making this transformation:

N !πm

(N +m)!√(

Nk

)√(Nj

) ⟨zj, zk⟩

=

∫Cm

(aw+b−bw+a

)j1 (aw+b−bw+a

)k1z j z

k(1 +

∣∣∣ aw+b−bw+a

∣∣∣2 + |z|2)N+m+1

dm(w)

| − bw + a|2m+2,

=

∫Cm

(aw + b)j1(aw + b)k1(−bw + a)(N−|j|)(−bw + a)(N−|k|)w2jdm(w)(| − bw + a|2 + |aw + b|2 + |w|2

)N+m+1,

=

∫Cm

(aw + b)j1(aw + b)k1(−bw + a)(N−|j|)(−bw + a)(N−|k|)w2jdm(w)

(1 + |w|2)N+m+1,

as |a|2 + |b|2 = 1.

=

⟨√(Nj

)(z1 − ζ)j1(1 + ζz)N−|j|z j

(1 + |ζ|2)N2

,

√(Nk

)(z1 − ζ)k1(1 + ζz)N−|k|zk

(1 + |ζ|2)N2

⟩,

12

We now show that both these bases are normalized by demonstrating

that the first one is. Explicitly, this is accomplished by switching to polar

coordinates and integrating by parts:∥∥∥∥∥√(

N

j

)zj

∥∥∥∥∥2

=1

πm

∫Cm

(N +m)!

j!(N − |j|)!|z1|2j1 ˜|z|

2jdm(z)

(1 + |z1|2 + |z|2)N+m+1,

=1

πm−1

∫Cm−1

∫ ∞

r=0

2(N +m)!

j!(N − |j|)!r2j1+1 ˜|z|

2jdrdm(z

(1 + |r|2 + |z|2)N+m+1),

=1

πm−1

∫Cn−1

∫ ∞

u=0

(N +m)!

j!(N − |j|)!uj1 ˜|z|

2jdudm(z)

(1 + u+ |z|2)N+m+1,

=1

πm−1

∫Cm−1

∫ ∞

u=0

(N +m− 1)!

(j1 − 1)!j!(N − |j|)!uj1−1 ˜|z|

2jdudm(z)

(1 + u+ |z|2)N+m,

...

=1

πm−1

∫Cm−1

∫ ∞

u=0

(N +m− j1)!

j!(N − |j|)!

˜|z|2jdm(z)

(1 + u+ |z|2)N+m−j1,

=1

πm−1

∫Cm−1

(N +m− j1 − 1)!

j!(N − |j|)!

˜|z|2jdm(z)

(1 + |z|2)N+m−j1−1,

...

= (N − |j| − 1)!

∫ ∞

u=0

du

(N − |j|)!(1 + u)N−|j|,

= 1,

Now let us recall our definition of a Gaussian random function, line (1):

ψα(z) =∑

αjψj(z).13

We define Gaussian entire flat functions to be Gaussian random functions

with ψj being the first basis we chose for H2(Cm, e−|z|2dm):

ψα(z) =∑

αjzj√j!, (2)

where αj are i.i.d. Gaussian random variables and we are using the stan-

dard multi-index notation. Prior to proving Theorem 2.3 this is an abstract

definition, but after we prove said theorem we will know that a Gaussian

entire flat function a.s. defines an entire function. Additionally, while this

definition appears to depend on the basis chosen, this is not the case as may

be seen by skipping ahead to Theorem 2.5, and the following corollary.

Likewise to define Gaussian random SU(m + 1) polynomials we choose

ψj to be a basis for PolyN , which gives

ψα,N(z) =N∑

|j|=0

αj

√(N

j

)zj, (3)

where αj are i.i.d. Gaussian random variables. As this sum is finite, we

see immediately that Gaussian random SU(m + 1) polynomials are almost

surely degree N polynomials.

2.2. The radius of convergence of a random power series. In this

section we will show that a Gaussian entire flat function, is in fact, a.s., an

entire function.

14

Proposition 2.2. Let α be a standard complex Gaussian Random Variable,

then: a-i) Prob(|α| ≥ λ) = e−λ2

a-ii) Prob(|α| ≤ λ) = 1− e−λ2 ∈ [λ

2

2, λ2], ifλ ≤ 1

a-iii) if λ ≥ 1 then Prob(|α| ≤ λ) ≥ 12.

b) If αjNm is a sequence of i.i.d. standard random variables then

lim sup |αj|1|j| = 1, almost surely.

Proof. of a)

Prob(|α| ≥ λ) =1

π

∫|α|>λ

e−|α|2

dm(α)

=

∫r>λ

2re−r2

dr

= e−λ2

Proof. of b) We will first prove that for all ε ∈ (0, 1], lim sup |αj|1|j| > 1− ε,

almost surely.

Prob(|αj| > (1− ε)|j|) = e(1−ε)2|j| ≥ e−(1−ε)2 . Thus the probability of

this event is greater than or equal to the event where in an infinite sequence

of coin flips involving a coin, which takes the value heads with probability

1e2−2ε , has at least one “heads” appear after the Rth flip, for any fixed R.

This event has probability one by the zero-one law.

We now prove that for all ε > 0, lim sup |αj|1|j| < 1 + 2ε, almost surely.

Prob(|αj| > (1 + ε)|j|) = e−(1+ε)2|j| . Therefore,

∑j∈Nn

Prob(|αj| > (1 + ε)|j|) =∑j∈Nm

e−(1+ε)2|j| <∞.

15

Hence,

Prob(lim sup |αj|1|j| > 1 + ε) = 0.

Theorem 2.3. If for all j ∈ N, ψj(z) ∈ O(Ω), such that for all compact

K ⊂ Ω, there exists an εK > 0 such that∑

(1 + εK)|j| maxz∈K

|ψj(z)| < ∞,

and if αj is a sequence of i.i.d. Gaussian random variables then for a.a.-

α,∑

αjψj(z), converges locally uniformly on B(0, r), and hence defines a

holomorphic function on Ω.

Proof. Let K be compact, and w ∈ K.∣∣∣∣∣∑j∈N

αjψj(w)

∣∣∣∣∣ ≤∑j∈N

|αj||ψj(w)| ≤∑j∈N

|αj|maxz∈K

|ψj(z)|. The result now fol-

lows by Proposition 2.2 and the ratio test.

Corollary 2.4. If ψα(z) is a Gaussian entire flat function, ψα(z) =∑αj

zj√j!

then ψα(z) is a.s. an entire holomorphic function.

Proof. For all r,∑j∈Nm

r2|j|

j!= emr

2

<∞, and hence the ψα is a.s. a holomor-

phic function on any polydisk by the previous theorem.

2.3. Invariance laws. We will now show that defining random holomor-

phic functions as∑

ψjαj, where ψj is any basis is a good definition, well

defined independent of basis.

Theorem 2.5. If ψjj∈Λ and φjj∈Λ are orthonormal bases for H a

Hilbert space of holomorphic functions, and if there exists ε > 0 such that16

∑(1 + ε)j|φj(z)|2 and

∑(1 + ε)j|ψj(z)|2, converge locally uniformly in a

domain R, and if αjj∈Λ are i.i.d. standard Gaussian random variables

then ∃βjj∈Λ i.i.d. standard Gaussian random variables s.t.

∀z ∈ R, ψα(z) = φβ(z), a.s.

Proof. An evaluation map is a linear map from H to C (since |ψj|2 con-

verges locally uniformly), and hence there exists aj such that∑

ajψj is

a representative for this fixed evaluation map:

∀f ∈ H, evaz0(f) = 〈f,∑

ajψj〉H = f(z0)

In particular,

evaz0(ψj) = aj = ψj(z0)

As both ψj and φj are orthonormal bases we have the following

relations:

〈φj, ψk〉 = uj,k, where ∀j,∑k

|uj,k|2 = 1, and ∀k,∑j

|uj,k|2 = 1

〈ψk, φj〉 = uj,k

(uj,k) = uk,j∑k

ul,kuk,j = δj,l

For all j we define a new Gaussian random variable by βj = 〈ψα, φj〉H , and

using these new Gaussian random variables we define a new Gaussian ran-

dom holomorphic function:

17

φβ =∑βjφj =

∑〈ψα, φj〉φj.

ψα(z0) = 〈ψα(z),∑

ajψj〉

= ψα(z0)

φβ(z0) = 〈∑k

〈ψα, φk〉φk,∑

ajψj〉

=∑j

aj〈∑k

〈ψα, φk〉φk, ψj〉

=∑j,k

aj〈ψα, φk〉 · 〈φk, ψj〉

=∑j,k,l

ajuk,lαl · uk,j

=∑j,k,l

ajul,k · uk,jαl

=∑j

ajαj

= ψα(z0)

Corollary 2.6. (Invariance laws)

1) If ψα,N is a Gaussian entire flat function then there exists βj a

sequence of i.i.d. standard Gaussian random variables such that

∑j∈Nm

αjzj√j!

= e−12|ζ|2−zζ

(∑j∈Nm

βj(z + ζ)j√

j!

)

2) If ψα,N is a degree N Gaussian random SU(m+1) polynomial then for

all ζ ∈ C1 there exists βj a sequence of i.i.d. standard Gaussian random18

variables such that

N∑|j|=0

αj

√(N

j

)zj =

N∑|j|=0

βj

√(Nj

)(z1 − ζ)j1(1 + ζz1)

N−|j|∏ zjkk

(1 + |ζ|2)N2

2.4. Zero sets of m variable holomorphic functions and currents.

In this section, we will examine the basic properties of zero sets. As a simple

example, consider the function f(z) = sin(z). As a set

Zf = f−1(0) = (2n+ 1)π, n ∈ N.

To this set we may associate a distribution:

Zf =∑n∈N

δnπ(z),

where δa is a Dirac delta function centered at a. This measure defines a

linear function on C∞c (C1) : if φ ∈ C∞

c (C1) then

〈Zf , φ〉 :=∑n∈Z

φ(nπ),

and as supp(φ) is compact this sum is finite.

We now generalize such a construction to m variables by defining cur-

rents and forms. These results and definitions can be found in ([Sch],[GH]).

Let K ⊂M , be compact. Let D(j,p)K be the (j,m) forms whose support is in

K. Let D(j,p)U = dir limK⊂UD(j,p)

K .

19

example 2.7. For M = Cm,

D(1,1)Cm =

∑i,j

φi,j(z)dzi ∧ dzj, φi,j ∈ C∞c

As long as supp(φ) = K ⊂ U, U a single coordinate chart, any (j, k)

form can be written in the above form.

Forms have an associated exterior derivative d : D(j,p)M → D(j+1,p)

M ⊕

D(j,p+1)M . This derivative can be “split” into holomorphic and anti-holomorphic

parts: d = ∂ + ∂. Explicitly, locally on generators of D(j,p)M , ∂fdzI ∧ dzJ =∑ ∂f

∂zidzi ∧ dzI ∧ dzJ . This gives the following identities:

1) d(α ∧ β) = (dα) ∧ β + (−1)deg(α)α ∧ (dβ)

2) dd = 0 = ∂∂ = ∂∂, d∂ = ∂∂, ∂∂ = (−1)∂∂

3) α ∈ D(m−1,m)Mm ⊕D(m,m−1)

Mm ⇒∫M

dα =

∫∂M

α = 0, (Stoke′s theorem)

Just as in 1 variable Zf was dual to C∞c , in m dimensions the zero set

is an analytic variety and this variety will be dual to (m− 1,m− 1) forms

(by taking restrictions). This brings up the definition of currents:

Definition 2.8. A current of bi-degree (j, p) is a linear map,

T : D(m−j,m−p)(M) → C that is continuous in the direct limit topology.

The set of all currents of degree (j, p) is thus Dual to D(m−j,m−p)M , and

will be denoted as: D′(j,p) :=(D(m−j,m−p)M

)′20

A (j, p) current is closely related to a (j, p) form as for a particular

ρ ∈ D(j,p)M ,

∫Mρ ∧ · ∈ D′(j,p) is a linear function on D(m−j,m−p)

M .

The zero sets of holomorphic functions are often given the added struc-

ture of divisors, which can be written as (1,1) currents. For zero sets, this

added structure is unique up to multiplication by nonzero holomorphic func-

tions. In one dimension this becomes: for f ∈ O(M1),

div0(f) =∑z∈Zf

nj[zj],

where nj is the order of f at zj, or the highest integer nj st m ≤ nj ⇒

f (m)(zj) = 0. For M1,D(0,0)(M) = C∞c (M), and as a current,

〈div0(f), φ〉 =∑z∈Zf

njφ(zj)

where the compactness of the support of φ guarantees that the former sum

is finite.

example 2.9. As a simple example in m variables consider: f(z) = z1, z ∈

Cm, Zf = 1 · [z1 = 0] = δz1=0dz1 ∧ dz1, the Dirac delta function, φ ∈

D(m−1,m−1)(C) where φ(z) =∑fi,j(z)dz1 ∧ . . . ∧ dzi ∧ . . . ∧ ˆdzj ∧ . . . ∧ dzn,

and

〈Zf , φ〉 =

∫supp(φ)

(δz1=0(z)dz1 ∧ dz1) ∧ φ(z)

=

∫supp(φ)∩z1=0

f1,1(0, z2, . . . , zn)dz2 ∧ dz2 ∧ . . . ∧ dzn ∧ dzn21

The above example is highly related to Dirac delta functions. Further,

log(|z|2), is a fundamental solution to the Laplace equation:

δ0 =i

2π∂∂ log |z|2

or in other words Zf(z)=z = δ0 = i2π∂∂ log |z|2 ∈ D′(1,1). In one dimension it

is very easy to prove a generalization of this:

i

2π∂∂ log |f(z)|2 = div0(f)

For Mm an m dimensional manifold, the situation is very similar, but

there are more details to verify. For f ∈ O(M), f : Mm → C, f−1(0)

is still a divisor, and is an m− 1 analytic variety. Hence the regular points

of Zf are a manifold, and it makes sense (taking restriction), to identify

forms in D(m−1,m−1)M with ones in D

(m−1,m−1)Zf,reg

. As Zf,reg is an n-1 complex

manifold,

∫Zf,reg

is a (1,1) current on M, which we will denote Zf,reg (abusing

notation).

Theorem 2.10. (Poincare-Lelong formula)

If f ∈ O(Mm), M an m complex manifold,

then Zf = i2π∂∂ log |f |2, as (1, 1) currents on M.

This theorem can be proved by using a theorem of Federer dealing with

integration of forms near singularities [Shi]. Computation rules for using

currents resemble those for differential forms, and the weak forms of the

standard properties of differential forms invariably hold:22

Proposition 2.11. (Some basic properties of currents)

1) T ∈ D′(p,q), φ ∈ D′(m−p−1,q), 〈∂T, φ〉 = (−1)p+q+1〈T, ∂φ〉

2) d2T = 0

3) d(T ∧ β) = d(T ) ∧ β + (−1)deg(T )T ∧ dβ

2.5. The expected zero set of a Gaussian random holomorphic

function. We will now prove a standard result wherein the expected zero

set of Gaussian random holomorphic functions is determined. Let us first

consider the one variable case. If one chooses a random polynomial of de-

gree 1, the zero set of this polynomial will consist of point charges in the

complex plane, counted with multiplicity. This polynomial occurs with zero

probability, and when we compute the average zero set these singular delta

measures will be smeared out into one which is absolutely continuous with

respect to Lebesque measure. The theorem below is a more precise and

further reaching statement of this heuristic reasoning.

Theorem 2.12. If E[|ψα(z)|2] = Π(z, z) converges locally uniformly in Ω

then

E[Zα⋂

Ω] = (i

2π∂∂ log ||ψ(z)||2`2)

∣∣Ω

= (i

2π∂∂ log(Π(z, z)))

∣∣Ω.

Many various forms of the following theorem have been proven, [EK],

[Kac] and [Sod]. For my purposes it is important that the proof is valid in

n dimensions, and for infinite sums. Many of the proofs resemble this one.

After a conversation with Steve Zelditch, I was able to simplify a previously23

complicated argument into the current form. This simplification is already

known to other researchers including Mikhail Sodin.

As a matter of notation we define Π(z, z) to be the Szego Kernel, which

for any orthonormal basis has the form

Π(z, w) =∑j∈N

ψj(z)ψj(w).

Proof. Let α ∈ Dm−1,m−1(Ω)

To simplify the notation, let α = φ dz2 ∧ dz2 ∧ . . . ∧ dzn ∧ dzn.

〈Zψα , α〉 = 〈 i2π∂∂ log(|ψα(z)|2), α〉

= 〈 i2π

log (|ψα(z)|2) , ∂∂α〉

= 〈 i2π

(log(Π(z, z)) + log

(|ψα(z)|2Π(z,z)

)), ∂∂α〉

Taking the expectation of both sides we compute:

E[〈Zψα , α〉] = 12π

∫α

∫z∈Ω

log(Π(z, z)) ∂2φ∂z1∂z1

dm(z) dν(α)

+ 12π

∫α

∫z∈Ω

log(|ψα(z)|2Π(z,z)

)∂2φ

∂z1∂z1dm(z) dν(α)

The first term is the desired result (which by assumption is integrable

and finite), while the second term will turn out to be zero. We first must

establish that it is in fact integrable:∫z∈Ω

∫α| log

(|ψα(z)|2Π(z,z)

)∂2φ

∂z1∂z1dν(α)| dm(z)

≤ c∫z∈K

∫α

∣∣∣log(|ψα(z)|2Π(z,z)

)dν(α)

∣∣∣ dm(z)

= c∫z∈K

∫α|log(|α′|2)| dν(α′) dm(z)

where α′ is a standard centered Gaussian (∀z). Integrability follows as:∫z∈Ω

∫α

∣∣∣∣log

(|ψα(z)|2

Π(z, z)

)∣∣∣∣ ≤ C

∫|x|<1

| log(x)|dm(x)+c

∫|x|>1

|xe−x2|dm(x) ≤ c

24

Finally,∫Ω

α ∧ E[Zψα ] =i

2π

∫Ω

∂∂α log(Π(z, z)) +

∫Ω

C∂2φ

∂zi∂zjdm(z)

=i

2π

∫Ω

∂∂α ∧ log(Π(z, z))

Corollary 2.13. Expected Zero set of specific gaussian random holomorphic

functions

1) For a Gaussian random entire flat function ψα,

E[Zψα ] =i

2π(dz1 ∧ dz1 + dz2 ∧ dz2 + . . .+ dzm ∧ dzm)

2) For a Gaussian random SU(m+ 1) polynomial ψα,N ,

E[Zψα,N] =

Ni

2π

(dz1 ∧ dz1 + dz2 ∧ dz2 + . . .+ dzm ∧ dzm)

(1 + |z|2)2

In theorem 2.12, we proved that the expected zero set is determined by

the variance of a of the random function when evaluated at a point. More

can be said:

Theorem 2.14. Let ψj and φj be two collections of holomorphic func-

tions defined on a domain Ω such that for all compact K ⊂ Ω, there exists

εK > 0 such that∑

(1 + ε)j maxK

|ψj(z)2| <∞,∑

(1 + ε)j maxK

|φj(z)2| <∞

and let αj be a sequence of i.i.d. Gaussian random variables.25

If E[Zψα ] = E[Zφα ] then there exists h ∈ O(Ω)x and a sequence of i.i.d.

standard random variables, βj, such that for all z ∈ Ω

ψα(z) = h(z)φβ(z)

This theorem is proven in one dimension by Sodin [Sod], and the same

proof works in m-dimensions.

Proof. Fix a domain Ω, and a collection of holomorphic functions, ψjj∈Λ,

such that∑

|ψj(z)|2 <∞. Let Π1(z, w) =∑j∈Λ

ψj(z)ψj(w).

By design, E[ψα(z)ψα(w)] = Π1(z, w) and by theorem 2.12 this deter-

mines the expected zero set.

Suppose E[φβ(z)φβ(z)] = Π2(z, z) has the same zero set. Hence, by

theorem 2.12:

∂∂ log Π1(z, z) = ∂∂ log Π2(z, z)

Or, in other words

logΠ1(z, z)

Π2(z, z)= H(z), ∂∂H(z) = 0

As ∂∂H(z) = 0, H(z) = 2Re(g(z)), and Π1(z, z) = eg(z)eg(z)Π2(z, z). While

a priori, this is only true on the diagonal, the values on the diagonal uniquely

determine Π1(z, w) in a neighborhood, hence

Π1(z, w) = eg(z)eg(w)Π2(z, w)26

As the Szego kernel is a trace class, and evaluation maps are linear

functionals we get that

φβ(z) = eg(z)ψα(z).

3. New results concerning large deviations in the

distribution of zeros of a Gaussian entire flat function.

3.1. An estimate for the growth rate of Gaussian entire flat func-

tions. In this section we begin working towards our main results. The first

step in this process will be to prove Lemma 3.2 which is interesting in and

of itself as it proves that Gaussian entire flat functions are of finite order 2,

a.s.

We will be need the following elementary estimate:

Proposition 3.1. If j ∈ N+,m then |j||j|jj ≤ m|j|

Proof. Let uk = jk|j| ≥ 0, hence

∑nk=1 uk = 1.

As∑ukδu−1

kis a probability measure:

m∑j=1

uk log

(1

uk

)≤ log

∑ 1

ukuk, by Jensen’s inequality.

= log(m)

Hence, m|j| ≥∏k

(uk)−|j|uk

= |j||j|jj

27

Let Mr,α = maxz∈B(0,r)

|ψα(z)|

Lemma 3.2. For all δ > 0 there exists Rm,δ and cδ,m > 0 such that for all

r > R, ∣∣∣∣ log(Mr,α)

r2− 1

2

∣∣∣∣ ≤ δ,

except for an event whose probability is less than e−cδ,mr2m+2

.

Proof. We will first prove that: Prob( log(Mr,α)

r2≥ 1

2+ δ ≤ e−cδ,1r

2m+2and

we will prove this by specifying an event Ωr of measure almost 1 where

Mr,α ≤ e(12+δ)r2 .

Let Ωr be the event where: i) |αj| ≤ eδr2

4 , |j| ≤ 2e ·m · r2

ii)|αj| ≤ 2|j|2 , |j| > 2e ·m · r2

Prob(Ωcr) ≤

∑|j|≤2e·mr2

Prob(|αj| > eδr2

4 ) +∑

|j|>2e·mr2Prob(|αj| > 2

|j|2 )

≤ cmr2me

(−e

δr2

2

)+

∑|j|>2e·mr2

e−2|j|

≤ e−ecr2

+ ce−2cr2

, ∀r > R0

≤ e−ecr2

We now have that Prob(Ωcr) < e−e

cr2

< e−cr2m+2

. It now remains for

us to show that ∀α ∈ Ωr,log |Mr,α|

r2≤ 1

2+ 1

2δ. For all α ∈ Ωr, and for all

z ∈ B(0, r),

Mr,α ≤ maxz∈B(0,r)

|j|≤4e·m( 12r2)∑

|j|=0

|αj||z|j√j!

+∑

|j|>4e·m( 12r2)

|αj||z|j√j!

= max

z∈B(0,r)

1∑+ max

z∈B(0,r)

2∑Using the Cauchy-Schwartz inequality:

28

maxz∈B(0,r)

1∑≤ (e

14δr2)√c(r2)m max

z∈B(0,r)

(∑j

|z2j|j!

) 12

≤ cmeδr2

4 rme12r2

≤ e(r2)( 1

2+ 1

3δ), ∀r > Rm.

Using Stirling’s Formula:

maxz∈B(0,r)

2∑≤ max

z∈B(0,r)

∑|j|>4e·mr2

(2)|j|2|zj|√j!

≤∑

|j|>4e·mr2(2)

|j|2

(|j|

4em

) |j|2 ∏

k

(e

jk

) jk2

≤ C, by Proposition 3.1.

Hence, for all α ∈ Ωr, log(Mr,α) ≤ (12

+ 12δ)r2, proving half of the result.

Let M ′r,α = max

z∈P (0,r)|ψα(z)|, where P (0, r) := z ∈ Cm : ∀i, |zi| < r

It now remains for me to show that for all δ > 0, and ∀r > Rm,δ:

Prob

(log(Mr,α)

r2≤ 1

2− δ

)≤ e−cδ,2r

2m+2

.

It suffices to prove this result only for small δ. However, we will actually

prove a stronger claim: that for all δ and for all r > Rδ,m there exists cδ,m

such that

Prob(M ′

r,α ≤m

2r2 − δr2

)< e−cδ,mr

2m+2

This is a stronger claim as: Mr,α ≥M ′1√mr,α⇒Mr,α <

(1

2− δ

)r2

⊂

M ′r,α <

(1

2− δ

)r2

and the desired probability estimate therefore will

hold by monotonicity.

29

Let us assume that

log(M ′r,α) ≤

(m2− δ)r2.

By Cauchy’s Integral Formula:∣∣∣∂jψα

∂zj

∣∣∣ (0) ≤ j!M ′r,αr

−|j|. Further, by direct

computation using line (1):∣∣∣∣∂jψα∂zj

∣∣∣∣ (0) = |αj|√j!

Therefore: |αj| ≤ cM ′r,α

√j!r−|j|, and using Sterling’s formula,

(j! ≤ e112

√2π√jjje−j), we get that for all k, jk 6= 0:

|αj| ≤ (2πe112 )

m2 (∏

k j14k )e(

m2−δ)r2+

∑ jk2

log(jk)−(|j|) log r− |j|2 .

The (2πe112 )

m2 j

14 term is will not matter in the end so we will focus in-

stead on the exponent, which we will now call A.

A =(m

2− δ)r2 − |j|

2+∑k

(jk2

log(jk)

)− (|j|) log(r)

=k=m∑k=1

(jk2

)((1− 2δ

m

)r2

jk− 1 + log(jk)− 2 log(r)

)If for all k, jk = r2 then A = −δr2. We now will generalize this observation

for jk ≈ r2 by letting jk = γkr2.

A =k=m∑k=1

(γkr

2

2

)((1− 2δ

m

)1

γk− 1 + log(γk)

)= −δr2 +mf(γk)

r2

2, where f(γk) = 1− γk + γk log(γk)

f(γk) = (1− γk)2 − (1− γk)

3 + o((1− γ)4) near 1.

30

Hence there exists ∆ such that ∀δ ≤ ∆ if γk ∈[1−

√δm, 1 +

√δm

]then

A ≤ −δr22

Therefore for j as above |αj| ≤ (2πe112 )

m2 (∏

k j14k )e−

δr2

2 ≤ cmrm2 e−

δr2

2 .

This holds true for all αj, j in terms of r. Specializing our work for large

r, we have that ∀ε > 0, there exists Rm,δ, such that for all r > Rm,δ, |αj| ≤

e−12(δ−ε)r2 . Here the factor of ε is used to compensate for the e

112

√2πj

14k

terms, which had been ignored.

We now prove the desired probability decay rate:

Prob(logM ′r,α ≤ (1

2− δ)r2)

≤ Prob(|αj| ≤ e−12(δ−ε)r2 , and jk ∈ [(1−

√δm

)r2, (1 +√

δm

)r2])

≤ (e−(δ−ε)r2)(2√

δmr2)m

= e−2m(1+o(δ))δm+2

2 r2m+2= e−c1,δr

2m+2, using the

independence of αj and Proposition 2.2.

Corollary 3.3. Let z0 ∈ B(0, r)\B(0, 12r).

For all δ > 0 there exists Rδ,m and cδ,m > 0 such that for all r > Rδ,m, there

exists ζ ∈ B(z0, δr) such that

log |ψα(ζ)| >(

1

2− 3δ

)|z0|2,

except for an event whose probability is ≤ e−cδ,mr2m+2

Proof. Without loss of generality assume that δ < 14. By Lemma 3.2:

Prob( maxz∈∂B(0,r)

log |ψα(z)| −1

2|z|2 ≤ −δr2) ≤ e−cδr

2m+2

.

31

By Corollary 2.6, we have that for z0 ∈ B(0, r)\B(0, 12r), z ∈ B(z0, δr):

Prob( maxz∈∂B(0,δr)

log |ψα(z − z0)| −1

2|z − z0|2 ≤ −δ(δr)2) ≤ e−cδr

2m+2

.

Hence, there exists cδ,m, Rδ,m and z ∈ B(z0, δr) such that log |ψα(z − z0)| −12|z − z0|2 ≥ −δ(δr)2, except for an event whose probability for r > Rδ,m is

less than e−cr2m+2

.

By hypothesis, |z0| ∈ [12r, r), hence |z − z0| ≤ δr ≤ 1

4r = r

212≤ 1

2|z0|.

Hence, |z0 − z|2 ≥ |z0|2 − δr2 ≥ |z0|2(1− 2δ).

log |ψα(z − z0)| ≥ 12|z − z0|2 − δ3r2 ≥ |z0|2 1

2(1− 2δ)2 − 4δ3|z0|2

≥ 12|z0|2 − 2δ|z0|2 − 1

4δ|z0|2

≥ 12|z0|2 − 3δ|z0|2

After setting ζ = z − z0 we have proven what we set out to prove.

Using that log maxB(0,r)

|ψα| is an increasing function in terms of r, we may

prove the following corollary:

Corollary 3.4. For all δ > 0

a) Prob

(limr→∞

(log maxz∈B(0,r) |ψα(z)|)− 12r2

r2/∈ [−δ, δ]

)= 0

b) Prob

(limr→∞

(log maxz∈B(0,r) |ψα(z)|)− 12r2

r26= 0

)= 0

This corollary has been proven by more direct methods, [SZ].

Proof. Part b follows immediately from part a, which we now prove.32

Let Eδ,R =

log maxB(0,R) |ψα(z)|− 12R2

R2 /∈ [−δ, δ]

, and let

Rt = r + δ(t+ 1)r, r > 0. Let st ∈ [Rt−1, Rt].

Claim: ∀t > Tδ, ∀st, Eδ,st ⊂ E 13δ,Rt

⋃E 1

3δ,Rt−1

Let Tδ = maxT1,δ, T2,δ, which may be specifically determined.

Case i: for α ∈ Eδ,st , log maxB(0,st)

|ψα| ≥1

2s2t + δs2

t

log maxB(0,Rt)

|ψα| ≥ 12s2t + δs2

t ,

≥ 12(1 + tδ)2r2 + δ(1 + tδ)2r2

> 12R2t + 1

3δR2

t , ∀t > M1,δ

Therefore, α ∈ E δ3,Rt

Case ii: for α ∈ Eδ,st , log maxB(0,st)

ψα ≤1

2s2t − δs2

t

log maxB(0,Rt−1)

|ψα| ≤ 12s2t − δs2

t

≤ 12(1 + (t− 1)δ)2r2 − δ(1 + tδ)2r2

≤ 12R2t−1 − 1

3δR2

t−1, ∀t > M2,δ

Therefore, α ∈ E δ3,Rt−1

. Hence, ∀t > Tδ and ∀s ∈ [Rt−1, Rt],

Eδ,s ⊂ E 13δ,Rt−1

∪ E 13δ,Rt

. Hence, Prob(⋃

s∈[Rt−1,Rt]Eδ,s

)≤ 2e−cδr

2m+2t2m+2,

and ∑t∈N

Prob

⋃s∈[Rt−1,Rt]

Eδ,s

=∑t∈N

e−cδt2m+2

<∞.

The result follows.

3.2. Large deviations in the value of the Nevanlinna characteristic

function of a Gaussian entire flat function. In order to prove Theorem33

1.1 we will need to estimate

∫Sr

log |ψα|dµr, where dµr is the rotationally

invariant Haar probability measure on the sphere Sr = ∂B(0, r). This will

be proved by approximating a surface integral using Riemann integration.

In order to establish notation I state the Poisson integral formula: for

ζ ∈ B(0, r), and h a harmonic function,

h(ζ) =

∫Sr

Pr(ζ, z)h(z)dµr(z),

where Pr is the Poisson kernel for B(0, r). For this notation, the Poisson

kernel is given by the equation:

Pr(ζ, z) = r2m−2 (r2 − |ζ|2)|ζ − z|2m

Lemma 3.5. There exists Rm and cm > 0 such that for all r > Rm,∫Sr

| log(|ψα|)|dµr(z) < (32m + 1)r2,

except for an event whose probability is less than or equal to e−cmr2m+2

.

Proof. By Lemma 3.2, with the exception of an event whose probability is

less than e−cmr2m+2

, there exists ζ0 ∈ ∂B(0, 12r) such that log(|ψα(ζ0)|) > 0.

Hence, ∫∂B(0,r)

Pr(ζ0, z) log(|ψα(z)|)dµr(z) ≥ log(|ψ(ζ0)|) ≥ 0.

Alternatively,∫∂(B(0,r))

Pr(ζ0, z) log−(|ψα(z)|) ≤∫∂(B(0,r))

Pr(ζ0, z) log+(|ψα(z)|)

34

Since ζ ∈ ∂B(0, 12r) and z ∈ ∂B(0, r), we have: r2

4≤ |z − ζ|2 ≤ 9

4r2.

Estimating the values of the Poisson Kernel for these values of z and ζ

yields:

1

3

(2

3

)2m−2

≤ Pr(ζ, z) ≤ (2)2m−23

Therefore, ∫Sr

log+(|ψα(z)|)dµr(z) ≤ logMr ≤(

1

2+ δ

)r2 ≤ r2,

except for an event whose probability is less than e−cmr2m+2

, by Lemma 3.2.

It remains to compute∫

log− |ψα| :∫∂(B(0,r))

P (ζ0, z) log+(|ψα(z)|) ≤ µr(Sr) log(Mr)3(2)2m−2

≤ 3(2)2m−2r2∫∂(B(0,r))

log−(|ψα(z)|)dµr(z) ≤ 1minz P (ζ0,z)

∫∂(B(0,r))

P (ζ0, z) log+(|ψα(z)|)

≤ 3(

32

)2m−2 ∫∂(B(0,r))

P (ζ0, z) log+(|ψα(z)|)

≤ 9(

32

)2m−2(2)2m−2r2

≤ 32mr2

The result now follows immediately.

In order to use Riemann integration to prove Lemma 3.9 we will need

to be able to choose “evenly” spaced points on the sphere. This choice will

be made according to the next lemma:

Lemma 3.6. (A partition of a Sphere)

For all t ∈ N+, let Q = (2m)t2m−1, then S2mr ⊆ R2m can be partitioned into

35

Q disjoint measurable sets Ir1 , Ir2 , . . . , IrQ such that

diam(Irj ) ≤√

2m− 1

tr =

cm

Q1

2m−1

r.

Proof. Surround Sr with 2m pieces of planes: P+,1, P+,2, . . . , P+,m, P−,1,

. . . , P−,m, where

P+,j = x ∈ R2m : ||x||L∞ = r, xj = r

P−,j = x ∈ R2m : ||x||L∞ = r, xj = −r

Subdivide each piece into t2m−1 identical closed 2m− 1 cubes, in the usual

way, and enumerate these sets R′1, . . . , R

′Q. These sets may intersect one

another on the boundary, which is a problem that may be fixed by letting

Rj = R′j\⋃k<j R

′k.

Let Irj = x ∈ Sr : λx ∈ Rj, λ > 0. Clearly, x ∈ Sr ⇒ ∃ ! j such that

x ∈ Irj .

Also, by design λ ≥ 1 and x, y ∈ Irj therefore,

d(x, y) ≤ d(λ1x, λ2y) ≤2

tr = diam(Rj).

For the following result note that the integration is with respect to w,

which is not the same variable of integration that is usually used in the

Poisson integral formula.

36

Lemma 3.7. For κ < 1, z ∈ ∂B(0, r)∫w∈Sm

κr

Pr(w, z)dµκr(w) = 1

Proof. If w ∈ S2mκr ⊆ R2m, then the Poisson Kernel can be rewritten as a

function of |z − w|, and as such ∀Υ ∈ Un(Rn), Pr(Υw,Υz) = Pr(w, z)

Let f(z) =∫w∈Sn

κrPr(w, z)dµκr(w)

f(z) =∫w∈Sm

κrPr(w, z)dµκr(w)

=∫w∈Sm

κrPr(Υw,Υz)dµκr(w), by the above work.

=∫w∈Sm

κrPr(Υw,Υz)dµκr(Υw), as dµκr is invariant under rotations.

=∫w∈Sn

κrPr(w,Υz)dµκr(w), by a change of coordinates.

= f(Υz)

Hence f(z) = c, ∀z ∈ SmrBy switching the order of integration we compute that:

1 =

∫w∈Sm

κr

∫z∈Sm

r

Pr(w, z)dµr(z)dµκr(w) = c

Lemma 3.8. If r > 0 and δ ∈ (0,min1r, 1), κ = 1 − δ

12(2m+2)(2m−1) , and

Iκrj Qj=1 forms a partition of the sphere of radius κr, and is chosen as per

lemma 3.6, and if for each j a point ζj is chosen such that d(ζj, Iκrj ) < δr

then

maxz∈∂(B(0,r))

∣∣∣∣∣Q∑j=1

µκr(Iκrj )(Pr(ζj, z)− 1)

∣∣∣∣∣ ≤ Cmδ1

2(2m−1)

Proof. Let µj = µκr(Iκrj )

37

∀z ∈ ∂B(0, r),∫ζ∈∂B(0,κr)

Pr(ζ, z)dµκr(ζ) = 1, by Lemma 3.7. Hence,

1 =

j=N∑j=1

µjPr(ζj, z) +

j=N∑j=1

∫ζ∈Iκr

j

(Pr(ζ, z)− Pr(ζj, z))dµκr(ζ).

|∑j=N

j=1 µj(Pr(ζj, z)− 1)| = |∑j=N

j=1

∫ζ∈Iκr

j(Pr(ζ, z)− Pr(ζj, z))dµκr(ζ)|

≤ maxj, ζ∈Iκr

j

|ζ − ζj| · maxw∈B(0,(κ+δ)r)\B(0,(κ−δ)r)

∣∣∣∣∂Pr(w, z)∂w

∣∣∣∣We now approximate the LHS:

∂Pr(w, z)

∂w= −r2m−2w|z − w|2 + (r2 − |w|2)m(z − w)

|z − w|2m+2

As |z| = r, and |w| = (1 − ε)r ∈ [(κ − δ)r, (κ + δ)r] this last line may be

approximated as:∣∣∣∣∂Pr(w, z)∂w

∣∣∣∣ ≤ 2 + 4εm

rε2m+2≤ cmrε2m+2

=cmrδ−

12(2m−1)

Using that maxζ |ζ − ζj| ≤ diam(Ij) + δr ≤ cδ1

2m−1 r+ δr ≤ c′rδ1

2m−1 , we

may finish the proof of the claim:∣∣∣∣∣j=N∑j=1

µj(Pr(ζj, z)− 1)

∣∣∣∣∣ ≤ Cδ1

2m−1 · δ−1

2(2m−1) = Cδ1

2(2m−1)

.

Now we are able to prove our final lemma.

38

Lemma 3.9. For all ∆ > 0, there exists Rm,∆ > 0 and c∆,m > 0 such that

for all r > Rm,∆

1

r2

∫z∈∂B(0,r)

log |ψα|dµr(z) ≥1

2−∆,

except for an event whose probability is less than e−cm,∆r2m+2

.

Proof. It suffices to prove the result for small ∆. Let am = 12(2m+2)(2m−1)

.

Set δ =

(1

λ∆

)( 1am

)

<1

6, λ > 0 to be determined later. Choose m ∈ N such

that 1(2m)m2m−1 ≤ δ, and for notational purposes let N = (2m)m2m−1. Also,

let κ = 1− δam .

Form a disjoint partition, Iκrj , of Sκr as in Lemma 3.6. In particular,

diam(Iκrj ) ≤ cδ1

2m−1 r.

Let µj = µκr(Iκrj ), which does not depend on r, and for all j fix a point

xj ∈ Iκrj .

By Corollary 3.3, for each j there exists a ζj ∈ B(xj, δr) such that

log(|ψα(ζj)|) >(

1

2− 3δ

)|xj|2 =

(1

2− 3δ

)κ2r2

except for N different events each of which has probability less than e−c′r2m+2

,

and thus the union of all of these also has probability less than e−cr2m+2

.

As we have the same estimate for each j for |ψα(ζj)|, and∑µj = 1 we

have:39

(1

2− 3δ

)(1− δam)2r2 ≤

N∑j=1

µj log(|ψα(ζj)|)

≤∫∂B(0,r)

(∑j

µjPr(ζj, z) log(|ψα(z)|)dµr(z)

)

=

∫∂(B(0,r))

(∑j

µj(Pr(ζj, z)− 1)

)log(|ψα(z)|)dµr(z)

+

∫∂(B(0,r))

log(|ψα(z)|)dµr(z)

Hence,∫∂B(0,r)

log(|ψα|)dµr

≥ (12− 3δ)(1− δam)2r2 −

∫| log |ψα||dµr ·maxz |

∑j µj(Pr(ζj, z)− 1)|

≥ (12− 3δ)(1− δam)r2 − (32m + 1)r2 · Cmδ

12(2m−1) ≥ 1

2r2 − λδamr2

by Lemma 3.5 and lemma 3.8.

This lemma gives an alternate proof for the growth rate of the charac-

teristic function. Let T (f, r) =∫Sr

log+ |f(z)|dµr(z), the Nevanlinna char-

acteristic function.

Corollary 3.10. For all δ ∈ (0, 13]

a) Prob

(limr→∞

(∫Sr

log |ψα|dµr)− 12r2

r2/∈ [−δ, δ]

)= 0

b) Prob

(limr→∞

(∫Sr

log |ψα|dµr)− 12r2

r26= 0

)= 0

c) Prob

(limr→∞

T (ψα, r)− 12r2

r26= 0

)= 0

40

As (∫Sr

log |ψα|dµr) is increasing the proof of Corollary 3.4 can be used

in conjunction with Lemma 3.9 to prove that ψα(z) is a.s. finite order 2. A

more elementary proof of this is already available [SZ].

3.3. Large deviations in the volume of the zero sets of Gaussian

entire flat functions, and a derivation of the order of the decay

of the hole probability. We will now be able to put the pieces together

to estimate the value of the unintegrated counting function of a Gaussian

entire flat function ψα(z).

Definition 3.11. For f ∈ O(B(0, r)), B(0, r) ⊂ Cm, the unintegrated

counting function,

nf (r) :=∫B(0,t)

⋂Zf

( i2π∂∂ log |z|2)m−1 =

∫B(0,t)

( i2π∂∂ log |z|2)m−1∧ i

2π∂∂ log |f |

The equivalence of these two definitions follows by the Poincare-Lelong

formula. The above form (( i2π∂∂ log |z|2)m−1) gives a projective volume, with

which it is more convenient to measure the zero set of a random function.

The Euclidean volume may be recovered as∫B(0,t)

⋂Zf

(i

2π∂∂ log |z|2)m−1 =

∫B(0,t)

⋂Zf

(i

2πt2∂∂|z|2)m−1.

Lemma 3.12. If f ∈ O(Br), and f(0) 6= 0 then

∫ t=R

t=r 6=0

dt

t

∫Bt

i

2π∂∂ log(|f |2) ∧ (

i

2π∂∂ log |z|2)m−1

=1

2

∫SR

log |f |2dµR −1

2

∫Sr

log |f |2dµr

41

This result is proven on pages 390-391 of Griffiths and Harris, [GH].

We may now prove one of our main theorems, Theorem 1.1:

Proof. (of theorem 1.1).

It suffices to prove the result for small δ.

As nψα(r) is increasing,

nψα(r) log(κ) ≤∫ t=κr

t=r

nψα(t)dt

t≤ nψα(κr) log(κ)

Let κ = 1+√δ. Except for an event whose probability is less than e−cr

2m+2,

we have:

nψα(r) log(κ) ≤∫ t=κr

t=r

nψα(t)dt

t

=

∫ t=κr

t=r

∫B(0,t)

i

2π∂∂ log |ψα(z)| ∧

(i

2π∂∂ log |z|2

)m−1dt

t

=1

2

∫Sκr

log |ψα(z)|dµκr −1

2

∫Sr

log |ψα(z)|dµr, by Lemma 3.12.

≤ 1

2

((1

2+ δ

)κ2r2 −

∫Sr

log |ψα(z)|dµr)

, by Lemma 3.2.

≤ 1

2

((1

2+ δ

)r2κ2 −

(1

2− δ

)r2

), by Lemma 3.9.

2nψα(r)

r2≤ 1

log(κ)

(κ2

(1

2+ δ

)−(

1

2− δ

))=

κ2 − 1

2 log(κ)+ δ

κ2 + 1

log(κ)≤ 1 + c

√δ.

42

We have just shown that:

Prob

(nψα(r)

r2≥ 1

2+ δ

)≤ e−cδr

2m+2

and now only half of the result remains to be proven. We now complete

the proof by noting that, except for an event whose probability is less than

e−cr2m+2

, we have that: nψα(r) log(κ) ≥∫ t=r

t=κ−1r

nψα(t)dt

t, by line (2).

=

∫ t=r

t=κ−1r

∫B(0,t)

i

2π∂∂ log |ψα(z)| ∧

(i

2π∂∂ log |z|2

)m−1dt

t

=1

2

∫Sr

log |ψα(z)|dµr −1

2

∫Sκ−1r

log |ψα(z)|dµκ−1r, by Lemma 3.12.

≥ 1

2

[(1

2− δ

)r2 −

∫Sκ−1r

log |ψα(z)|dµκ−1r

], by Lemma 3.9.

≥ 1

2

[(1

2− δ

)r2 −

(1

2+ δ

)r2κ−2

], by Lemma 3.2.

2nψα(r)

r2≥ 1

log(κ)

((1

2− δ

)− (

1

2+ δ)κ−2

)=

1− κ−2

2 log(κ)− δ

1 + κ−2

log(κ)≥ 1− 2

√δ

We have now finished the proof by showing that:

Prob

(nψα(r)

r2≤ 1

2− δ

)≤ e−cδr

2m+2

Using this estimate for the typical measure of the zero set of a random

function we get an upper bound for the hole probability, and a lower bound

with the same order of decay is easy to prove:

43

Proof. (of theorem 1.2).

The upper estimate follows by the previous theorem, as if there is a hole

in a ball of radius r then nψα(r) = 0, and this can only occur for an event

with probability less than e−cr2m+2

. Therefore it suffices to show that the

event where there is a hole in the ball of radius r contains an event whose

probability is larger than e−cr2m+2

. We now design such a set. Let Ωr be the

event where:

i) |α0| ≥ Em + 1,

ii) |αj| ≤ e−(1+ m2

)r2 , ∀j : 1 ≤ |j| ≤ d24mr2e = d(m · 2 · 12)r2e

iii) |αj| ≤ 2|j|2 , |j| > d24mr2e ≥ 24mr2

Prob(|αj| ≤ e−(1+m2

)r2) ≥ 12(e−(1+ m

2)r2)2 = 1

2e−(2+m)r2 , by Proposition 2.2

#j ∈ Nm|1 ≤ |j| ≤ d24mr2e = ((d24mr2e+m

m

)) ≈ cr2m

Hence, Prob(Ωr) ≥ C(e−cmr2m+2

) ≥ e−cr2m+2

, by independence and Propo-

sition 2.2. It now suffices to show that for all α ∈ Ωr, and for all z ∈

B(0, r), ψα(z) 6= 0.

|ψα(z)| ≥ |α0| −|j|≤d24mr2e∑

|j|=1

|αj|r|j|√j!−

∑|j|>d24mr2e

|αj|r|j|√j!

= |α0| −1∑−

2∑1∑

≤ e−(1+m2

)r2|j|≤d24mr2e∑

|j|=1

r|j|√j!

≤ e−(1+m2

)r2√

(24mr2 + 1)m√

(erm), by Cauchy-Schwarz inequality.

≤ Cmrme−r

2 ≤ ce−0.9r2 < 12

for r > Rm

44

2∑≤

∑|j|>24mr2

2|j|2

(|j|

24m

) |j|2 1√

j!, as r <

√|j|

24m

≤ c∑

|j|>24mr2

2|j|2

(|j|

24m

) |j|2k=m∏k=1

(e

jk

) jk2

, by Stirling’s formula

= c∑

|j|>24mr2

(|j|)|j|2(∏k=m

k=1 jjk2k

)m

|j|2

( e12

) |j|2

≤ c∑|j|>1

(1

4

) |j|2

, by Proposition 2.2.

≤ c∑l>1

(1

2

)llm ≤ Em

Hence, |ψα(z)| ≥ Em + 1−1∑−

2∑≥ 1

2

4. New results concerning large deviations of the

distribution of zeros of a Gaussian random SU(m+ 1)

polynomial

4.1. Large deviations of the maximum value of a random SU(m+1)

polynomial on a ball of radius r. Our goal will now be to estimate

maxB(0,r)

log |ψα,N |. This next lemma is key as it states that the maximum of

the norm of a random SU(m + 1) polynomial on the ball of radius r tends

to not be too far from its expected value.

Lemma 4.1. For all δ ∈ (0, 1), and for all r > 0 there exists ar,δ,m > 0 and

Nδ,m such that for all N > Nδ,m

45

maxB(0,r)

|ψα,N(z)| ∈[(1 + r2)

N2 (1− δ)

N2 , (1 + r2)

N2 (1 + δ)

N2

],

except for an event whose probability is less than e−ar,δNm+1

.

Proof. We will first prove a sharper decay estimate for the probability of

the event where a random SU(m + 1) polynomial takes on large values in

the ball of radius r:

Prob

(maxB(0,r)

|ψα,N(z)| > (1 + r2)N2 (1 + δ)

N2 )< e−cmN

2m

.

To do this we consider the event ΩN := ∀j, |αj| ≤ Nm, the complement

of which has probability ≤ (N+1)me−N2m

, by Proposition 2.2. For α ∈ ΩN ,

maxz∈B(0,r)

|ψα,N(z)| = maxz∈B(0,r)

∣∣∣∣∣∑αj

(N

j

) 12

(z)j

∣∣∣∣∣≤ max

z∈B(0,r)

∑|αj|(N

j

) 12

|z|j

≤ maxz∈B(0,r)

Nm(N + 1)m2 (1 +

∑|zi|2)

N2 ,

by the Schwartz inequality.

= Nm(N + 1)m2 (1 + r2)

N2

For all δ > 0, limN→∞

cmNm (N + 1)

m2

(1 + δ)N2

= 0, therefore there exists N ′δ,m,1 > 0

such that if N > N ′δ,m,1 then cmN

m(N + 1)m2 < (1 + δ)

N2 . Hence, for

N > N ′δ,m,1

maxz∈B(0,r)

|ψα,N(z)| < (1 + δ)N2 (1 + r2)

N2 .

In other words, if N > N ′δ,m,1 then

maxB(0,r)

|ψα,N(z)| > (1 + r2)N2 (1 + δ)

N2

⊂ Ωc

N

46

and thus there exists N ′δ,m > N ′

δ,m,1 such that this event has probability less

than or equal to (N + 1)me−N2m

< e−12N2m

, for all N > N ′δ,m. This decay

rate is independent of δ and r, and the estimate for the order of the decay

of this probability could be improved upon.

We complete the proof by showing that:

Prob

(maxB(0,r)

|ψα,N(z)| < (1 + r2)N2 (1− δ)

N2 )< e−ar,δN

m+1

.

This will be done when we prove the following claim concerning a poly-

disk, P (0, 1√mr) := z ∈ Cm : |z1| < 1√

mr, |z2| < 1√

mr, . . . , |zm| < 1√

mr:

Prob

(maxP (0,r)

|ψα,N(z)| < (1 +mr2)N2 (1− δ)

N2 )< e−ar,δN

m+1

.

This second claim is stronger as maxP (0, 1√

mr)|ψα,N(z)| ≤ max

B(0,r)|ψα,N(z)|.

Consider the event where

M = maxP (0,r)

|ψα,N(z)| < (1 +mr2)N2 (1− δ)

N2 .

We will show that this event can only occur if certain Gaussian random

variables, αj, obey the inequality |αj| < e−cδ,mN , where cδ,m > 0. Further

we will show that this occurs whenever j is in a certain cube. This will give

us the desired decay rate for the probability.

The Cauchy estimates for a holomorphic function state that:

|ψ(j)α,N(0)| ≤ j!

M

r|j|.

47

By differentiating a random SU(m+ 1) polynomial of degree N and evalu-

ating at zero we compute that

ψ(j)α,N(0) =

√(N

j

)j!αj.

Combining this with Stirling’s formula:

√2πj1j

j11 e

−j1 < j1! <√

2πj1jj11 e

−j1e112

we get that:

|αj| ≤ (1 +mr2)N2 (1− δ)

N2

r|j|√(

Nj

)≤ e

m12

+m2

log(2π)

(1 +mr2)N2 (1− δ)

N2 (N − |j|) 1

2(N−|j|+ 1

2)∏

(jk)jk+1

22

r|j|NN+1

22

≤ e

m12

+m2

log(2π)Nm4 ·

((1 +mr2)

N2 (1− δ)

N2 (N − |j|) 1

2(N−|j|)∏(jk)

jk2

r|j|NN2

)For the time being we focus on the term in parenthesis in the previous line

which we call A. Writing j as j = (jk) = (xkN), xk ∈ (0, 1), we now have:

A = (1− δ)N2

((1 +mr2)

r2|x| (1− |x|)(1−|x|)m∏k=1

(xk)xk

)N2

If for all k ∈ 1, 2, . . . ,m, xk = r2

1+mr2then A = (1−δ)N

2 , which inspires

the following claim:

Claim: Let sr,m = 12m(1+mr2)

min r2, 1.

If for each k ∈ 1, 2, . . . ,m, xk ∈[

r2

1+mr2− sr,mδ,

r2

1+mr2

]⊂ (0, 1), and thus

|x| < 1 then

48

(1 +mr2)( xr2

)x(1− |x|)(1−|x|) < (1− δ)−

12 .

Proof: We begin by setting xk = (1−∆k)r2

1+mr2and ∆ =

∑∆k. There-

fore

∆k ∈[0,

1

2mmin

1,

1

r2

δ

]and ∆ ∈

[0,

1

2min

1,

1

r2

δ

].

Thus, 1− |x| = 1+∆r2

1+mr2, and from this we compute that:

(1 +mr2)

r2|x| (1− |x|)(1−|x|)(xk)xk =

1 +mr2

r2|x|

(1 + ∆r2

1 +mr2

)1−|x|

·(

r2

1 +mr2

)|x|∏(1−∆k)

xk

=(1 + ∆r2

)1−|x|∏(1−∆k)

xk

≤(1 + ∆r2

)≤ 1 +

1

2δ

≤(

1

1− δ

) 12

.

Proving the claim.

Therefore if for each k, xk ∈[

r2

1 +mr2− sr,mδ,

r2

1 +mr2

], then A < (1−

δ)N4 . This in turn guarantees that |αj| < e

m12

+m2

log(2π)Nm4 (1 − δ)

N4 . The

probability this occurs for a single αj is less than or equal to

(e

n12

+n2

log(2π)Nm2 (1− δ)

N4

)2

.

Thus the probability this occurs for all αj, jk ∈[( r2

1+mr2− sr,mδ)N, (

r2

1+mr2)N],

is less than or equal to

(e

m12

+m2

log(2π)Nm4 (1− δ)

N4

)2(bNsr,mδc)m

.

49

Hence, there exists ar,δ,m > 0 and N ′′δ,m such that for all N > N ′′

δ,m,

(e

m12

+m2

log(2π)Nm4 (1− δ)

N4

)2(bNsr,mδc)m

< e−ar,δ,mNm+1

The result follows after setting Nδ,m = maxN ′δ,m, N

′′δ,m

A nice application of this lemma is the following:

Lemma 4.2. For all ∆ ∈ (0, 1) and a ∈ Cn\0 there exists N∆,|a|,m and

c∆,|a|,m > 0, such that if N > N∆,|a|,m then

maxz∈B(0,∆)

|ψα,N(z − a)| > (1 + |a|2)N2 (1−∆)

N2 ,

except for an event whose probability is less than e−c∆,|a|,mNm+1

.

Proof. As Gaussian random SU(m+ 1) polynomials are rotationally invari-

ant, as a random process, with out loss of generality we assume that a is of

the form: a = (ζ1, 0, . . . , 0).

Let δ = ∆2+2|ζ|+2|ζ|2)

.

By Lemma 4.1 and line (2), there exists c∆,|a|,m > 0 and N∆,|a|,m such

that if N > N∆,|a|,m then, except for an event whose probability is less than

e−c∆,|a|Nm+1

,50

(1− δ)N2 ≤

maxB(0,δ) |ψα,N(z)|(1 + δ)

N2

= max∂B(0,δ)

∣∣∣∣∣∑α′j

√(N

j

)(z1 − ζ1)

j1(1 + ζ1z1)N−|j|

∏(zk)

jk

∣∣∣∣∣(1 + |ζ1|2)

N2 (1 + δ2)

N2

.

In order to simplify this previous line, let φ(z) = ( z1−ζ11+ζ1z1

, z21+ζ1z1

, . . . , zm

1+ζ1z1),

so that we may rewrite the previous equation as:

(1− δ)N2 ≤

(max∂B(0,δ)

|1 + ζ1z1|N

(1 + |ζ1|2)N2 (1 + δ2)

N2

)(maxB(0,δ)

|ψα′,N(φ(z))|)

≤

((1 + |ζ1|δ)N

((1 + |ζ1|)2(1 + δ2))N2

)(max

B(−ζ1,(2+2|ζ1|2)δ)|ψα′,N(z)|

),

as the image of φ |B(0,δ)⊂ B(−ζ1, (2 + 2|ζ1|2)δ), since:

maxz∈∂B(0,δ)

∣∣∣∣ z1 − ζ

1 + ζz1

+ ζ

∣∣∣∣ = maxz∈∂B(0,δ)

∣∣∣∣z1 − ζ + ζ + z1|ζ|2

1 + ζz

∣∣∣∣= δ max

z∈∂B(0,δ)

∣∣∣∣(−1− ζ2)

1 + ζz1

∣∣∣∣≤ 2δ(|ζ|2 + 1)

Rearranging the previous sets of equations we get the result:

maxB(0,∆)

|ψα′,N(z − ζ1)| ≥ (1 + |ζ1|2)N2 (1 + δ2)

N2

(1 + |ζ1|δ)N· (1− δ)

N2

≥ (1 + |ζ1|2)N2 (1− (2 + 2|ζ1|)δ)

N2

≥ (1−∆)N2 (1 + |ζ1|2)

N2

4.2. Large deviations in the value of the Nevanlinna characteristic

function of a Gaussian SU(m+1) polynomial of degree N . The goal

of this section will be to estimate∫Sr

log |ψα,N(z)|dµr(z),51

where dµr(z) is the normalized rotationally invariant volume measure of

the sphere of radius r, Sr = ∂B(0, r), which we will accomplish when we

prove lemma 4.4, using the same techniques as in [ST2], and as were used

in section 2.

Lemma 4.3. For all r > 0 there exists cm,r, and Nm such that for all

N > Nm,∫Sr

| log(|ψα,N(z)|)|dµr(z) ≤(

32m

2+

1

2

)N log

((2)(1 + r2)

)except for an event whose probability is < e−cm,rNm+1

.

Proof. By Lemma 4.1, there exists cm,r, and Nm,r such that if N > Nm,r

then, with the exception of an event whose probability is less than e−cm,rNm+1,

there exists ζ0 ∈ ∂B(0, 12r) such that log(|ψα,N(ζ0)|) > 0. This also implies

that: ∫z∈Sr

Pr(ζ0, z) log(|ψα,N(z)|)dµr(z) ≥ log(|ψ(ζ0)|) ≥ 0,

Where Pr is the Poisson kernel for the sphere of radius r (whose area is

normalized to 1): Pr(ζ, z) = r2m−2 r2−|ζ2|

|z−ζ|2m . Hence,∫z∈Sr

Pr(ζ0, z) log−(|ψα,N(z)|)dµr(z) ≤∫z∈Sr

P (ζ0, z) log+(|ψα,N(z)|)dµr(z)

Now given the event where

log maxB(0,r)

|ψα(z)| <N

2log((2)(1 + r2)

),

52

(whose complement for N > Nm,r has probability less than e−cm,rNm+1), we

may estimate that∫z∈Sr

log+(|ψα,N(z)|)dµr(z) ≤N

2log((2)(1 + r2)

).

Since ζ0 ∈ ∂B(0, 12r) and z = reiθ, we have: r2

4≤ |z− ζ0|2 ≤ 9

4r2. Hence, by

using the formula for the Poisson Kernel,

22m−2

32m−1≤ P (ζ0, z) ≤ 3 · 22m−2.

Putting the pieces together proves the result:

∫z∈Sr

Pr(ζ0, z) log+(|ψα,N(z)|)dµr ≤ 3 · 22m−3N log(2(1 + r2)

)∫z∈Sr

log−(|ψα,N(z)|)dµr(z) ≤ 1

minP (ζ0, z)

∫z∈Sr

Pr(ζ0, z) log+(|ψα,N |)dµr(z)

≤ 32m

2N log

(2(1 + r2)

)

We now arrive at the main result of this section:

Lemma 4.4. For all r > 0 and for all ∆ ∈ (0, 1) there exists c∆,r,m > 0

and N∆,r,m such that for all N > N∆,r,m,∫z∈Sr

log(|ψα,N(z)|)dµr(z) >N

2log((1 + r2)(1−∆)

),

except for an event whose probability is less than e−c∆,r,mNm+1

.

Proof. It suffices to prove this result for small ∆. Set δ = 3−4m∆4m. Let

s = d1δe, let Q = (2m)s2m−1, and let κ = 1− δ

14m .

53

In lemma 3.6 it was shown that the sphere of radius κr may be parti-

tioned into Q measurable disjoint sets Iκr1 , Iκr2 , . . . , IκrQ such that

diam(Iκrj) ≤√

2m− 1

sκr =

cm

Q1

2m−1

κr.

We choose such a partition and then we choose a ζj within δr < 1 of Iκrj

such that

log(|ψα,N(ζj)|) >N

2log((1 + κ2r2)(1− δr)

),

for which, by Lemma 4.2, there exists c′∆,r and N ′∆,r such that if N >

N∆,r then the probability that this does not occur is less than e−c′δ,rN

m+1

.

Therefore there exists c∆,r > 0 and N∆,r such that if N > N∆ the union of

these m events has probability less than or equal to(2m

⌈1

δ

⌉2m−1)e−c

′δ,rN

m+1

< e−cδ ,rNm+1

.

Let µk = µκr(Iκrk ). As Iκr1 , Iκr2 , . . . , IκrQ form a partition of Sκr,

∑k

µk = 1.

We now turn to investigating the average of log |ψα,N(z)| on the sphere

of radius r by using Riemann integration and line (3):

N

2log((1 +

1

2κ2r2) (1− δ)) ≤

k=Q∑k=1

µk log |ψα,N(ζk)|

≤∫z∈Sr

(∑k

µkPr(ζk, z) log(|ψα,N(z)|)dµr(z)

)

=

∫z∈Sr

(∑k

µk(Pr(ζk, z)− 1)

)log(|ψα,N(z)|)dµr(z)

+

∫z∈Sr

log(|ψα,N(z)|)dµr(z)

This will simplify to:

54

∫log(|ψα,N(z)|)dµr(z) ≥ N

2log ((1 + κ2r2)(1− δr))

−(∫| log |ψα,N(z)||dµr(z))

· maxz∈Sr

|∑k

µk(Pr(ζk, z)− 1)|

In Lemma 3.8, it was computed that in exactly this situation that:

max|z|=r

∣∣∣∣∣∑k

µk(Pr(ζk, z)− 1)

∣∣∣∣∣ ≤ Cmδ1

2(2m−1)

Hence by Lemma 4.3 and line (4), there exists cδ,r,m > 0 and Nδ,r,m such

that if N > Nδ,r,m, except for an event of probability < e−cδ,r,mNm+1

:∫log(|ψα,N |)dµ(z) ≥ N

2log((1 + κ2r2)(1− δr)

)−CmN log

(2(1 + r2)

)δ

12(2m−1) ,

=N

2log((1 + r2 − 2δ

14m r2 + δ

12m r2)(1− δr)

)−CmN log

(2(1 + r2)

)δ

12(2m−1) ,

=N

2log[(1 + r2)− 2δ

14m r2

+δ1

2mO(r2 + δ4m−14m r3 + δ

2m−22m r)]

−CmN log(2(1 + r2)

)δ

12(2m−1) ,

≥ N

2log((1 + r2)(1− 3δ

14m )), for sufficiently small δ,

which is independent of N . The proof is thus completed by choosing suffi-

ciently small ∆ so that the previous line holds, (and δr < 1) .

4.3. Main results concerning random SU(m + 1) polynomials. We

will now be able to estimate the value of the unitegrated counting function

for a random SU(m + 1) polynomial, ψα,N , and by doing so we will prove55

one of our two main theorems, Theorem 1.3:

Proof. (of theorem 1.3). It suffices to prove the result for small ∆. Let

δ = ∆2

4< 1. Let κ = 1 +

√δ = 1 + ∆

2. As nψα,N

(r) is increasing,

nψα,N(r) log(κ) ≤

∫ t=κr

t=r

nψα,N(t)dt

t≤ nψα,N

(κr) log(κ)

There exists cδ,r,m > 0 and Nδ,r,m such that for all N > Nδ,r,m, except

for an event of probability ≤ e−cδ,r,mNm+1

, we get that:

nψα,N(r) log(κ) ≤

∫Sκr

log |ψα,N(z)|dµκr(z)−∫Sr

log |ψα,N(z)|dµr(z)

≤ N

2

(log((1 + κ2r2)(1 + δ)

)−∫Sr

log |ψα(reiθ)|dµr)

,

by Lemma 4.1.

≤ N

2

(log((1 + κ2r2)(1 + δ)

)− log

((1 + r2)(1− δ)

)),

by Lemma 4.4.

≤ N

2

(2√δr2 + δr2 + 2δ + 2δr2

(1 + r2)− 2δr4

(1 + r2)2+O(δ

32 )

),

Therefore,

nψα,N(r) ≤ N

(r2 + 1

2

√δr2 +

√δ +

√δr2

(1 + r2)−

√δr4

(1 + r2)2+O(δ)

)

·

(1 +

√δ

2+O(δ)

)≤ Nr2

1 + r2+ 3N

√δ +O(δ)

This proves the probability estimate when the value of the unintegrated

counting function nψα,N(r) is significantly above its typical value. We now

modify the above the argument to finish the proof. There exists cδ,r,m and

Nδ,r,m such that if N > Nδ,r,m, then except for an event whose probability56

is less than e−cδ,r,mNm+1

, the following inequalities hold:

nψα,N(r) log(κ) ≥

∫Sr

log |ψα,N(z)|dµr(z)−∫Sκ−1r

log |ψα,N(z)|dµκ−1r(z)

≥ N

2

(log((1 + r2)(1− δ)

)−∫Sκ−1r

log |ψα(reiθ)|dµ

),

by Lemma 4.4.

≥ N

2

(log((1 + r2)(1− δ)

)− log

((1 + κ−2r2)(1 + δ)

)),

by Lemma 4.1.

≥ N

2log(1− δ)

−N2

log

(1− 2

√δr2

1 + r2+

δr2

1 + r2+ δ +O(δ

32 )

).

Therefore,

nψα,N(r) ≥ N

2

(−√δ +

2r2

1 + r2−√δr2

1 + r2−√δ +

2√δr4

(1 + r2)2+O(δ)

)

·

(1 +

√δ

2+O(δ)

)≥ Nr2

1 + r2− 4N

√δ +O(δ).

We have just implicitly proven an upper bound on the order of the decay

of the hole probability. We will now compute the lower bound to finish the

proof theorem 1.4.

Proof. (of theorem 1.4) The desired upper bound for the order of the decay

of the hole probability is a consequence of the previous theorem.

We must still prove the lower bound for the order of the decay of the hole

probability, and we start this by considering the event, Ω which consists of

αj where:57

|α0| ≥ 1

|αj| <(N

j

)−12

N−mr−|j|.

If α ∈ Ω, then |α0| >N∑

|j|>0

|αj|(N

j

) 12

rj. Hence for all z ∈ B(0, r), ψα,N(z) 6=

0 ⇒ Ω ⊂ HoleN,r. A lower bound for the probability of Ω will thus give

a lower bound for the probability of HoleN,r. First we restrict ourselves to

considering the Gaussian random variables, αj, whose indices, j, satisfy the

equation:(Nj

)−1N−2mr−2|j| ≤ 1. For these j

Prob

(|αj| <

(N

j

)−12

N−mr−|j|

)≥ 1

2

1

N2m(Nj

)r2|j|

,

by Proposition 2.2.

=1

2

(N − |j|)!j!N2mN !

r−2|j|

≥ (2π)m2

N2m2(m+ 1)N+m2

r−2|j|e112

≥ e−(N+m2

) log(m+1)+cm

·e−|j| log(r)−2m log(N)

≥ e−(N+m2

) log(m+1)+cm

·e−|j| log(maxr,1)−2m log(N)

≥ e−cm,rN

Whereas if for the index j,(Nj

)−1N−2mr−2|j| > 1 then

Prob

(|αj| <

(N

j

)−12

N−mr−|j|

)≥ Prob ( |αj| < 1)

>1

2

≥ e−N log(2)

58

Further, Prob(|α0| > N) = e−1. Hence, Prob(Ω) ≥ e−cr,mNm+1

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60

Vita

Scott Zrebiec was born in July of 1978, and was raised in Wanaque, New Jersey

and Boulder, Colorado. In 2000, he received a Bachelors of Arts degree from CU,

Boulder. He enrolled at Johns Hopkins University in the Fall of 2001. He defended his

thesis on March 21, 2007.

63

Department of Mathematics, Johns Hopkins University, Baltimore, MD

21218, USA

E-mail address: szrebiec@math.jhu.edu

61