Polyanalytic Bergman Kernels

ANTTI HAIMI

Doctoral Thesis

Stockholm, Sweden 2013

ISSN 1401-2278

ISRN KTH/MAT/DA 13/02-SE ISBN 978-91-7501-742-6

Institutionen för Matematik 100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik tisdagen den 28 maj 2013 kl 13.00 i sal F3, Kungliga Tekniska högskolan, Lindstedts- vägen 26, Stockholm.

Antti Haimi, 2013 c

Tryck: Universitetsservice US AB

Abstract

The thesis consists of three articles concerning reproducing kernels of weighted spaces of polyanalytic functions on the complex plane. In the first paper, we study spaces of polyanalytic polynomials equipped with a Gaus- sian weight. In the remaining two papers, more general weight functions are considered. More precisely, we provide two methods to compute asymptotic expansions for the kernels near the diagonal and then apply the techniques to get estimates for reproducing kernels of polyanalytic polynomial spaces equipped with rather general weight functions.

Sammanfattning

Denna avhandling innehåller tre artiklar relaterade till reproducerande kärnor för viktade rum av polyanalytiska funktioner. I artikel A studeras polyanalytiska polynom i planet med en gaussisk vikt. I artiklarna B och C betraktas mer allmänna vikter. Mer precist presenterar vi två metoder att beräkna asymptotiska utvecklingar av kärnorna nära diagonalen och tillämpar sedan resultaten för att studera asymptotik för reproducerande kärnor av viktade polynomrum.

Contents v

Acknowledgements vii

Part I: Introduction and summary

1 Introduction 3

2 Summary of results 15

References 21

Part II: Scientific papers Paper A

The Polyanalytic Ginibre Ensembles 32 pages.

Paper B

Asymptotic expansion of Polyanalytic Bergman Kernels 41 pages.

Paper C

Bulk asymptotics for Polyanalytic Correlation Kernels 38 pages.

v

First, I would like to express my sincere gratitude to my supervisor Håkan Heden- malm for suggesting the topic of this thesis, helping with smaller and larger technical problems and sharing his mathematical ideas generously.

I also want thank Serguei Shimorin for comments regarding the third article of this thesis and Michael Benedicks for reading through the manuscript.

I also wish to thank the mathematics department at KTH for the nice working environment and my fellow graduate students for the great time we have spent together.

I am very grateful to my parents Kari and Marjatta, my sisters Sanna and Jonna, and my friends in Finland who have been supporting me all the way.

The thesis is dedicated to my wife Kerttu whose love and engourament have been essential.

vii

Introduction and summary

1

Introduction

Notation

We will write ∂X and int(X) for the boundary and the interior of a subset X of the complex plane C. By 1

we mean the characteristic function of the set X. We let

dA(z) = π

dxdy, where z = x + iy ∈ C,

be a suitably normalized area measure in C, and use the standard Wirtinger deriva- tives

∂

:=

(∂

− i∂

), ¯ ∂

:=

(∂

+ i∂

).

Will omit the subscripts if there is no risk of confusion. We write ∆ = ∂ ¯ ∂, which is one quarter of the usual Laplacian. We write D for the open unit disk, and more generally D(z, r) for the disk with center z and radius r. Given a Lebesgue mea- surable function w : C → R, we denote by L

(w) the space of measurable functions C → C which are square-integrable with respect to the measure w(z)dA(z).

Spaces of polyanalytic functions

The thesis is concerned with Hilbert spaces formed by so called polyanalytic func- tions. Because these functions arise as generalizations of analytic functions, we will start by defining the latter in a way which is suitable for our purposes. Given an open set Ω in C, a continuous function u : Ω → C is called analytic (or holomorphic) if it satisfies the the differential equation

∂u(z) = 0 ¯

in the sense of distributions. Analytic functions are central objects in complex analysis and their properties have been extensively studied by now.

Polyanalytic functions can be defined similarly but as solutions to differential equations of higher order. A continuous function u is called q-analytic if it solves the differential equation

∂ ¯

u(z) = 0, z ∈ Ω

3

for some integer q ≥ 1. A function which is q-analytic for some q ≥ 1 is called polyanalytic. By solving a ¯ ∂-equation repeatedly, we see that any q-analytic function can be written as

u(z) =

q−1

X

r=0

¯

z

u

(z), (1.1)

where the functions u

, . . . , u

are analytic.

The decomposition (1.1) implies that function theory of polyanalytic functions is very much related to that of analytic functions. A general reference in this topic is the book of Balk [5]. In this thesis, we will not concentrate on properties of individual polyanalytic functions very much– instead, we will be studying certain Hilbert spaces formed by them.

Let Q be a real-valued and continuous function on Ω.We define P A

(Ω) := {u continuous : ¯ ∂

u = 0, kuk

:=

Z

Ω

|u(z)|

e

dA(z) < ∞}.

Most of the time, we will use the simplified notation P A

for P A

(Ω). The function Q will be called the weight in the sequel. We will sometimes abuse termi- nology and refer to the function e

as the weight – hopefully, this does not lead to confusion.

We equip the spaces P A

(Ω) with the inner product hu, vi :=

Z

Ω

u(z)v(z)e

dA(z).

Proposition 1.1. The spaces P A

(Ω) are closed in L

(Ω, e

dA).

Proof. For z ∈ Ω and 1 ≤ k ≤ q − 1, let us define the linear functionals F

: P A

(Ω) → C, u 7→ ¯ ∂

u(z).

These mappings are always bounded, as is shown by the following computation.

We will be using that

is the fundamental solution of the operator ¯ ∂

(this follows easily from Leibniz rule for distributions and the fact that 1/(πw) is the fundamental solution of ¯ ∂). We take an arbitrary smooth cut-off function χ which is supported in a compact subset of Ω and which equals 1 in a neighborhood of z.

∂ ¯

u(z) = (−1)

(q − 1)!

Z

Ω

∂ ¯

∂ ¯

u(w)χ(w)
(w − z)

q−1

w − z dA(w)

= (−1)

(q − 1)!

Z

Ω q−k−1

X

j=0

∂ ¯

u(w) ¯ ∂

χ(w) (w − z)

w − z dA(w)

= Z

supp ¯∂χ

u(w)H

(w, z)dA(w), (1.2)

where

H

(w, z) := 1 (q − 1)!

q−k−1

X

j=0

(−1)

∂ ¯



∂ ¯

χ(w) (w − z)

w − z



(1.3)

is C

and compactly supported in Ω. An application of Cauchy-Schwarz inequality then gives

| ¯ ∂

u(z)| =     Z

Ω

u(w)e

H

(w, z)e

dA(w)    

≤

 Z

Ω

|u(w)|

e

dA(w)



 Z

Ω

|H

(w, z)|

e

dA(w)



. (1.4) This shows that the mappings F

are bounded. Because any reasonable choice of χ makes the norms locally uniform in z, it follows that spaces P A

(Ω) are closed in L

Ω, e

dA
.

This proposition implies that P A

(Ω) is always a Hilbert space. Because in the proof we showed that point evaluation functionals F

are bounded, we can deduce from Riesz representation theorem that for each z ∈ Ω there exists K

∈ P A

(Ω) which satisfies

u(z) = hu, K

i. (1.5)

It is customary to write K

(w) = K

(w, z). The function K

which satisfies (1.5) is called the reproducing kernel of the space P A

(Ω). It follows easily that K

(z, w) = K

(w, z). The reproducing kernel is unique and given any orthonormal basis {e

} of P A

(Ω), it can be written as

K

(z, w) = X

j

e

(z)e

(w),

where the sum converges uniformly on all compact subsets of Ω × Ω. We will call functions K

polyanalytic Bergman kernels, because K

is usually called the Bergman kernel (with respect to the weight e

and domain Ω).

Bergman spaces of polyanalytic functions have been studied previously in two model cases. In the first one, Ω = D and Q = 0, and in the second Ω = C and Q(z) = |z|

. In both cases, the reproducing kernels have been concretely calculated (see Koshelev [25] and Vasilevski [36], respectively). In the second example, the spaces are closely related eigenspaces of a certain magnetic Hamiltonian – this connection will be explained in more detail when we will discuss the results of paper A.

It should be noted that given general Ω and Q, it can be very difficult to compute

K

exactly. The purpose of this thesis is to provide methods for estimating these

kernels. Instead of considering the issue with a fixed weight Q, we will use a family of weights mQ, where m is a large positive scaling parameter. We will see, for example, that given some extra assumptions on Q, asymptotic expansions for K

can be obtained near the diagonal as m → ∞.

Apart from P A

, we will be interested in the polynomial spaces P A

:= span

{¯ z

z

: 0 ≤ r ≤ q − 1, 0 ≤ j ≤ n} ∩ P A

( C).

These spaces are of course finite-dimensional and therefore possess reproducing kernels K

. We will study these kernels as the analytic degree n tends to infinity. As above, we will also here replace Q by mQ, and let m tend to infinity. It turns out that it is natural to require that the two parameters m and n are rather close to each other, for example |m − n| = O(1).

Analytic Bergman kernels

Analytic Bergman kernels were introduced by Stefan Bergman in the model case Ω = D and Q = 0. Later, they have been studied extensively in several complex variables literature. On strongly pseudoconvex domains in C

equipped with a constant weight, this line of investigation was initiated by Hörmander [24]. The most well-known result in this setting is due to Fefferman [16], who obtained an asymptotic expansion for the Bergman kernel near the boundary of the domain and then used it in the study of biholomorphic mappings.

Bergman kernels can also be studied on more general complex manifolds, not just C

. Here it is natural to consider holomorphic sections of a line bundle instead of holomorphic functions. In this setting, weights of the form e

arise naturally.

To make matters more precise, suppose we are given a complex manifold X and a holomorphic line bundle L over X supplied with an Hermitian metric h. In our simplified setting above, h = e

is a local representation of the metric h. Next, if m is a positive integer, the weight e

appears as the local metric associated with the tensor power L

of the line bundle L. The reason to consider tensor powers is that the initial line bundle L might admit only very few holomorphic sections.

A central result concerning Bergman kernels on line bundles can be attributed to Tian [35], Yau [37], Zelditch [39] and Catlin [10]. We will formulate it only in our one complex variable terminology. Let ˜ K

be the q-analytic Bergman kernel with respect to the measure e

∆QdA on a domain Ω, i.e. reproducing kernel of the Hilbert space

P A g

(Ω) := {u continuous : ¯ ∂

u = 0, Z

Ω

|u(w)|

e

∆Q(w)dA(w) < ∞}.

We assume here that Q is strictly subharmonic, i. e. ∆Q > 0. The result states that there exists an asymptotic expansion of the kernel ˜ K

near the diagonal as m → ∞:

K ˜

(z, w) ∼ (mb

(z, w) + b

(z, w) + 1

m b

(z, w) + . . . )e

,

where Q(z, w) is a local polarization of the weight Q, i.e. a function which is holomorphic in (z, ¯ w) near the diagonal {(z, z) ∈ C

} and satisfies Q(z, z) = Q(z).

The coefficient functions here b

are holomorphic in (z, ¯ w). By now, there are several approaches to show this, see [27], [11], [8] and [31].

In paper B, we will consider similar asymptotic expansions for kernels K

. It seems that there is no essential difference between working with K

and K ˜

. For example, we can observe that ˜ K

= K

, where Q

(z) :=

Q(z) −

log ∆Q(z); this could provide one possibility of transferring asymptotic expansion results from kernels K

to kernels ˜ K

.

Determinantal point processes

We will in a moment use polyanalytic polynomial kernels K

to define so called determinantal point processes in the complex plane - this will provide a probabilistic intepretation for our kernel estimates. Determinantal processes were introduced by Macchi [28] to study systems of non-interacting fermions in quantum mechanics.

We will now quickly review the intuition behind this concept.

Let (X, µ) be a measure space. In quantum mechanics, the position of a par- ticle is represented by a wave function ψ ∈ L

(X, µ), which is assumed to satisfy R

Λ

|ψ|

dµ = 1. The quantity |ψ(x)|

then gives the probability density of finding the particle at the position x. Now, consider a linearly independent set ψ

, . . . , ψ

of individual particle wave functions and the N -body system consisting of these par- ticles. A natural question is how to construct the wave function Ψ

(x

, . . . , x

) for the composite system. If all the particles are fermions, then according to Pauli exclusion principle, the composite wave function has to be anti-symmetric with respect to interchange of two wave functions. This suggests that we could set

Ψ

(x

, . . . , x

) = c

det ψ

(x

) 

j,k=1

, (1.6)

where c

is a normalization constant. This choice clearly satisfies the desired property. One should however keep in mind that the composite wave function has such a simple form only if the fermions are assumed to be non-interacting - in the presence of an interaction, the wave function would be more complicated.

The probability amplitude corresponding to this wave function is

|Ψ

|

= c

det ψ

(x

) 

j,k=1

· det ψ

(x

) 

j,k=1

= c

det K

(x

, x

) 

j,k=1

, (1.7) where K

(x, y) = P

j=1

ψ

(x)ψ

(y). We can here assume that the functions ψ

are orthonormal: if they were not in the first place, we could orthonormalize them with Gram-Schmidt process and this would only change the normalization constant.

Given this assumption, we have c

=

; this follows easily after expanding the two determinants in the product and using orthonormality of the wave functions.

Notice that K

is the integral kernel of the orthogonal projection to the subspace

spanned by the functions ψ

, . . . ψ

.

Motivated by this discussion, we say that a probability measure dP (x

, . . . , x

) := Λ

(x

, . . . , x

)dµ(x

) . . . dµ(x

) on X

is a finite determinantal projection process if

Λ

(x

, . . . , x

) = 1

N ! det K

(x

, x

) 

,

where K

is an integral kernel of a projection to finite-dimensional subspace of L

(X, µ). It is customary to identify all copies of the space X and the points x

and think the process as a random configuration of N unlabelled points in X.

Even though the probability density Λ

is completely explicit, for many pur- poses, one needs to understand marginal densities of the process. Recall that for 1 ≤ k ≤ N , the k’th marginal density is obtained from Λ

by integrating out the variables with index ≥ k + 1:

Λ

(x

, . . . , x

) :=

Z

X^{N −k}

Λ

(x

, . . . , x

)dµ(x

) . . . dµ(x

).

An important property of determinantal processes is that marginal densities can also be written by a similar determinant involving the kernel K

(see [19]):

Λ

(x

, . . . , x

) = (N − k)!

N ! det K

(x

, x

) 

i,j=1

.

Because we would like to think the points of the process to be unlabelled, it is often more natural to use so called k-point intensity functions instead. The definition is as follows:

Γ

(x

, . . . , x

) := N !

(N − k)! Λ

(x

, . . . , x

) = det K

(x

, x

) 

. For any measurable subset A ⊂ X, the intensity functions satisfy

Z

A

Γ

(x

, . . . , x

)dµ(x

) . . . dµ(x

)

= E](j

, . . . , j

) ∈ perm(k, n) : (x

, . . . x

) ∈ A, (1.8) Here, E denotes the expectation operator, ] is the counting measure and perm(k, N) is the set of permutations of (1, . . . , N ) of length k. The one-point intensity Γ

is particularly important, because integrating it over a set gives the expected number of points in that set.

Finite determinantal projection processes are a subclass of more general deter-

minantal point processes, which can have varying or infinite number of points. We

are mostly interested in processes with only finite number of points, but for some

purposes, it is also useful to have the more general definition. We shall present that

now.

Let (X, µ) be a metric measure space and let M(X) denote the complete and separable metric space of bounded Borel measures. For our purposes, it is enough to take X to be a subset of C and dµ = ω(z)dA(z) for some positive and continuous function ω. A point process is a random locally finite integer valued measure Λ in M(Λ). Alternatively, we can view Λ as a random discrete subset of X. A point process is called simple if it almost surely assigns at most measure 1 to singletons.

A simple point process Λ is said to possess a k-point intensity function ρ

if

E X

x1,...,x_k∈Λ

f (x

, . . . , x

) = Z

X^k

f (x

, . . . , x

)ρ

(z

, . . . , z

)dµ(x

) . . . dµ(x

), .

where f is a bounded continuous function and the sum on the left hand side is over all k-tuples of distinct points in Λ. We should point out that there is no guarantee that intensity functions exist for a general point process.

A simple point process is called determinantal if there exist intensity functions ρ

for all k and they satisfy

ρ

(x

, . . . , x

) = det K(x

, x

) 

for some function K : X × X → C. It can be checked that finite determinantal projection processes are determinantal processes in this wider sense.

To the converse direction, one can ask whether a given kernel K defines a determinantal point process. This is answered by the following theorem, which in the form stated here can be found in [22].

Theorem 1.2 (Macchi [28], Soshnikov [33]). Let K be Hermitian, i.e. K(x, y) = K(y, x), positive definite and locally square-integrable. Let K define a self-adjoint integral operator K on L

(X, µ) which is locally trace class. Then, K defines a determinantal point process if and only if the eigenvalues of K lie on the interval [0, 1].

Now, take an open set Ω in C and let K

denote the reproducing kernel of P A

(Ω). By taking X = Ω, dµ = e

dA, we can check that the orthogo- nal projection K

defined by the kernel K

satisfies the conditions of the theorem. We conclude that K

defines a determinantal point process on Ω.

As for the polynomial kernels K

, we will use the notation Λ

(z

, . . . , z

) := 1

(nq)! det K

(z

, z

) 

nq×nq

e

P

j=1Q(zj)

(1.9) for the associated probability density. We need to require here that m, Q and n are such that the space P A

is nq-dimensional.

Weighted potential theory

To discuss previous results on analytic polynomial kernels, we need to recall some

facts from weighted potential theory. Let us assume that the weight Q is C

-regular

in C and satisfies the growth condition

Q(z) ≥ (1 + ) log |z|

, |z| > C (1.10) for some constants C,  > 0. We define N

and N

to be the sets

N

:= w ∈ C : ∆Q(w) > 0 , N

:= w ∈ C : ∆Q(w) ≥ 0 .

The equilibrium weight b Q is defined as the largest subharmonic function in C which is ≤ Q everywhere and has the growth bound

Q(z) = log |z| b

+ O(1), as |z| → +∞.

The function b Q is then subharmonic and it follows from general theory for obstacle problems that it is C

regular (see [20]). There is also an elementary proof of this fact due to Berman [6]. The coincidence set of the obstacle problem is

S := {Q(z) = b Q(z)}.

The set S is compact and because b Q is subharmonic, we have S ⊂ N

. It is known that b Q is harmonic on C\S. The set S is a central object in weighted potential theory as it arises as the support of the unique solution to the energy minimization problem

min

I(σ), where

I(σ) := 1 2

Z

C²

log 1

|z − w|

dσ(z)dσ(w) + Z

C

Q(w)dσ(w). (1.11) The infimum is taken over all compactly supported Borel probability measures.

Existence and uniqueness in this problem is due to Frostman. For more details, see [34]. Note that the functional (1.11) coincides with the standard logarithmic energy in the special case Q = 0. The solution to the energy minimization problem can be written as (see [20] or the earlier preprint [19])

b σ = ∆ b QdA = ∆Q1

dA.

The measure b σ will be referred to as the equilibrium measure.

We also need to consider a discrete version of the energy (1.11) . Replacing Q by mQ as before, we define

I

:= 1

n(n − 1) E

(z

, . . . , z

), (1.12) where

E

(z

, . . . , z

) := 1 2

X

j6=k

log 1

|z

− z

|

+ m

n

X

j=1

Q(z

). (1.13)

A point configuration z

, . . . , z

that minimizes the functional I

(or equiva- lently E

) is called a Fekete configuration of length n. If (z

, . . . , z

) denotes a Fekete configuration of length n, it is well-known that the probability mea- sures

P

j=1

δ

converge to σ in the weak-* sense of measures as m → ∞ and b m = n + o(n). We also have convergence of the energies:

I

(z

, . . . z

) → I( b σ).

For the proof in the case n = m, see [34] p. 145.

Bergman kernels for analytic polynomials

Kernels K

and associated probability densities Λ

were studied by Ameur, Hedenmalm and Makarov in [2], [3] and [4]. There are three possible interpretations for these point processes: in terms of Coulomb gas, free fermions or eigenvalues of random normal matrices.

As for the Coulomb gas model, we first rewrite (see [19] proposition 2.1) Λ

(z

, . . . , z

) = 1

Z

|∆(z

, . . . , z

)|

e

P

j=1Q(zj)

= 1 Z

e

, (1.14) where Z

is a normalization constant, ∆(z

, . . . , z

) = Π

(z

− z

) is the Vander- monde determinant and E

is the energy defined in (1.13). The last form of the probability density in (1.14) can be interpreted as a Gibbs measure in the sense of statistical mechanics. By this we mean a probability measure on C

of the form

1

Z ˜

e

dA(z

) . . . dA(z

),

where z

, . . . , z

are particles in C, E(z

, . . . , z

) is the energy of the configuration, T is the temperature of the system and ˜ Z

is a normalization constant. We con- clude that the determinantal point process given by Λ

describes a system of n negatively charged particles within an external field Q so that the interaction between the particles is logarithmic and temperature is fixed (T = 1).

For the free fermion picture, we use the fact that the functions

z

e

, j ≥ 0 (1.15)

are ground states of a Hamiltonian operator describing an electron in the complex

plane within a magnetic field that is perpendicular to the plane and of strength

which is proportional to m (see [38]). We now form a composite wave function

Ψ

of the form (1.6) from the n first eigenfunctions in (1.15). Here, we take the

reference measure dµ to be the normalized area measure dA. In view of (1.14)

and the previous discussion on determinantal point processes, we can view the

probability density Λ

as a quantum mechanical system of n non-interacting electrons of the type described.

The random matrix interpretation of Λ

arises as follows. We equip the set of n × n normal matrices with the probability distribution

1

Z

e

dM,

where dM is the volume form on the set of normal matrices induced by the standard metric on C

and Z

is a normalization constant. The weight Q is as before, but it is here lifted to act on normal matrices via functional calculus. It is well-known that the associated eigenvalue distribution is exactly Λ

.

We refer the interested reader to [38] for details concerning these three inter- pretations.

We assume from now on that Q is C

-smooth and satisfies the growth condition (1.10). As m, n → +∞ while n = m + o(n), Hedenmalm and Makarov ([19], [20]) showed, building on the work of Johansson [23], that

(n − k)!

n! Γ

dA

→ d σ b

(1.16)

in the weak-* sense of measures for all fixed k.

Here, the notation µ

stands for the k’th tensor power of the measure µ. As the authors showed in [20], this result implies that for any bounded continuous function g, we have the convergence

1 n

n

X

j=1

g(λ

) → Z

S

g(w)∆Q(w)dA(w), n, m → ∞, n = m + o(n) (1.17)

in distribution. The intuitive interpretation of this is that the points from the determinantal process defined via K

tend to accumulate on S with density

∆Q as m, n → +∞ while n = m + o(n). The set S was called droplet for this reason.

The convergence in (1.17) might not come as a surprise. Namely, if we recall the interpretation of Λ

as a Gibbs measure, one can view this as a random analogue of the above-mentioned result concerning convergence of Fekete configu- rations to the equilibrium measure.

Let us write bulk for the interior of the set S ∩ N

. The results of [3] show that for a bulk point z and an n-tuple (z

, . . . , z

) picked from

Λ

(z

, . . . , z

)dA(z

) . . . dA(z

), the local blow-up processes at z with coordinates

ξ

:= m

[∆Q(z)]

(z

− z)

1This result is just a special case of their theorem, which holds for Coulomb gas models in arbitrary temperatures.

converge to the the determinantal process defined by the reproducing kernel of the Bargmann-Fock space P A

( C) as m, n → +∞ while n = m + o(1). If we write e Γ

for the k-point intensity of the blow-up process, this means that

e Γ

(ξ

, . . . , ξ

) → det e



i,j=1

e

P

i=1|ξi|²

, n, m → ∞, n = m + o(1) (1.18) for all k and (ξ

, . . . , ξ

) ∈ C

.

This fact could be interpreted as a universality result in the spirit of random matrix theory and related fields. In general, universality means that there exists a scaling limit which does not depend on particularities of the model (see [13] for more discussion). As an example of this phenomenon, the reader could think of the classical central limit theorem, where a sum of i.i.d. random variables is (after suitable normalizations) seen to converge to a normal distribution. In particular, the limiting distribution is independent of the distribution of the individual random variables. Analogously, in our case the limiting process is the same for all weights Q.

One can also formulate a universality result more directly in terms of the kernel K

. To do this, let us define the Berezin density centered at z as

B

(w) := |K

(w, z)|

K

(z, z) e

. The main theorem in [2] was as follows:

Theorem 1.3 (Ameur, Hedenmalm, Makarov). Fix z ∈ intS ∩ N

and suppose that Q is real-analytic in some neighborhood of z. Then

1

m∆Q(z) B

(z + ξ

p∆Q(z)m ) → e

, n, m → ∞, m = n + o(1), where the convergence holds in L

C
.

It should be mentioned that Berman [7] proved a similar result independently, also in a higher-dimensional setting.

In paper C, we will generalize a slight reformulation of theorem 1.3 to the context

of more general polyanalytic polynomial kernels K

.

Summary of results

Paper A

In this paper, we study reproducing kernels K

for large n and m. When q = 1, the associated determinantal processes, defined by (1.9), are usually referred to as Ginibre ensembles. For general q, we call these processes polyanalytic Ginibre ensembles.

Polyanalytic Ginibre ensembles admit a physical interpretation in terms of free fermions. Recall the formula (1.7) for a determinantal process with a finite number of points. In this paper, we take the wavefunctions to be of the special form ψ(z) = u(z)e

, where u is an eigenfunction of the differential operator (densely defined on L

( C, e

))

∆ = −∂ ˜

∂ ¯

+ m¯ z ¯ ∂

.

This operator represents after suitable normalizations a single electron in the com- plex plane in a uniform magnetic field which is perpendicular to the plane. The eigenspaces for these operators are of the form (see [32], [30])

δP A

( C) := P A

( C) P A

( C).

Here, the symbol “ ” means taking the orthogonal difference of the two subspaces.

These eigenspaces are often referred to as Landau levels (in the literature, the term

“Landau level” can also refer to the eigenvalue).

We will denote by u

, j = 1, ..., n, some orthonormal basis of the space δP A

:= P A

P A

and write ψ

(z) := u

(z)e

for the corresponding wavefunctions. We now form the wavefunction for a system where there are n fermions at each of the q first Landau levels:

Ψ

(z) := c

det[ψ

(z

)]

, 1 ≤ i, j ≤ n, 1 ≤ r, s ≤ q,

where c

is a normalization constant. Here, we write z = (z

, ..., z

, ..., z

) for an nq-tuple of complex numbers. For physical reasons, it is natural to assume here that n ∼ m; this corresponds to each Landau level being completely filled.

15

Figure 1.1: The polyanalytic Ginibre process with the kernel Km,n,q with m = n = 61 and q = 3. The simulation is based on the algorithm by Hough, Krishnapur, Peres, and Virág [22].

The corresponding probability amplitude is 1

Z

|Ψ

(z)|

= 1

(nq)! det K

(z

, z

) 

nq×nq

, e

P

1≤r,i≤n|zr,i|²

, where Z

is another normalization constant. We conclude the polyanalytic Ginibre ensembles model systems of free fermions in a constant magnetic field so that all Landau levels below a given level are filled. There is no special signifigance in the choice of having exactly the same number of particles in each Landau level below the cut-off level – other configurations can be analyzed in a similar manner.

We will state our results in terms of the Berezin density and measure, which are defined by

B

(w) := |K

(w, z)|

K

(z, z) e

and

dB

(w) := B

(w)dA(w).

We find that the macroscopic behavior of polyanalytic Ginibre ensembles is similar to that of standard Ginibre ensembles (i.e. the special case q = 1). As m, n → +∞ while n = m + o(1), we have that

dB

→ dδ

for, z ∈ D, dB

→ dω

for z ∈ C\D,

where ω

is the harmonic measure of z with respect to the exterior disk C\D. Here, we use the notation

dB

(w) := B

(w)dA(w), B

(w) := |K

(z, w)|

K

(z, z) e

, for the corresponding Berezin measure and Berezin density. At microscopic length scales, behavior of the Berezin measure dB

is different for each q. In terms of the blow-up Berezin density

B ˆ

(ξ) = m

B

(z + m

ξ),

we have the following asymptotics as m, n → +∞ while n = m + O(1):

B ˆ

(ξ) −→ q

L

(|ξ|

)

e

for z ∈ D, (2.1) where L

denotes the associated Laguerre polynomial of degree q − 1 with pa- rameter 1.

We also investigate the local behaviour of the Berezin transform when |z| = 1, i.e., when z is on the boundary of the bulk. Using the same blow-up scale as with an interior point, we show that the blow-up Berezin density ˆ B

(ξ) tends to a limit which can be expressed in terms of Hermite polynomials.

Paper B

In this paper a general method to obtain asymptotic expansions for kernels K

on general domains Ω is presented for the case q = 2. Polyanalytic functions with q = 2 will be called bianalytic in the sequel.

We will assume that Q ∈ C

(Ω) and

∆Q(z) > 0 and sup

Ω

1

∆Q ∆ log 1

∆Q < +∞.

Then for any z

∈ Ω and small enough r > 0, we show that there exist approximate bianalytic Bergman kernels of the form:

K

(z, w) := m

b

(z, w) + mb

(z, w) + · · · + m

b

(z, w)e

, where b

, b

, b

, . . . are defined on D(z

, r) × D(z

, r) and bianalytic in (z, ¯ w).

They satisfy

 K

(z, w) − K

(z, w) 

e

= O m

, m → ∞, where the constant of the error is allowed to depend on z

, Q and r. An algorithm is presented to compute the coeffient functions b

, and the two first ones are com- puted as an example. The result is proved by extending the technique of Berman, Berndtsson and Sjöstrand [8] from the case analytic case q = 1.

We also suggest possible ways to use K

to define certain metrics on the

domain Ω.

Paper C

Here we study the polynomial kernels K

for large values of parameters m and n, mainly in the special case q = 2. As in the analytic case, we assume that Q is C

-smooth and satisfies the growth condition

Q(z) ≥ (1 + ) log |z|

+ O(1), for some C,  > 0 as |z| → ∞.

In terms of the one-point function Γ

(z) = K

(z, z)e

, it is shown that for any open neighborhood A of S, we have the convergence

Z

C\A

K

(z, z)e

dA(z) → 0

as m → ∞ and n ≤ m. This result is the same as in the analytic case q = 1.

The probabilistic interpretation in terms of Λ

is that the expected number of points in C\A tends to 0 as m → ∞ and n ≤ m.

In the bulk S ∩ N

, more detailed information is provided by the following theorem, which states that the scaling limits obtained for polyanalytic Ginibre ensembles are universal (i.e. independent of the weight Q) in the case q = 2. We will also discuss how to generalize the proof for q > 2.

Theorem 2.1. Set q = 2. Fix z

∈ intS ∩ N

and M > 0. Assume that Q is real-analytic in a neighborhood of z

. We define the rescaled coordinates

z := z

+ ξ

p∆Q(z

)m w := z

+ λ p∆Q(z

)m . Then, there exists a number m

such that for all m ≥ m

, we have

1 m∆Q(z

)

 K

(z, w) 

e

= |L

(|ξ − λ|

)|e

+ O m

, (2.2) as m → ∞ and n ≥ m − M . The convergence is uniform for compact subsets of (ξ, λ) ∈ C

. The constant of the error term may depend on Q, z

and r.

It can be observed that this theorem is a slight formulation of a bulk scaling limit obtained for the quadratic weight |z|

in paper A.

As an ingredient of the proof, we present another algorithm to compute asymp-

totic expansion of polyanalytic Bergman kernels. This approach is more elementary

than the previous one in the sense that a certain technique from microlocal analysis

used by Berman, Berndtsson and Sjöstrand is not needed. Our algorithm actually

provides an alternative way to obtain asymptotic expansions already in the analytic

case – however, it should be emphasized that the results of Berman-Berndtsson- Sjöstrand are proved on more general complex manifolds, whereas we are working only in subsets of the complex plane.

We also study off-diagonal decay of the kernels K

in the bulk. The theo- rem reads as follows:

Theorem 2.2. Suppose that Q is C

-smooth. Fix a compact set K in the interior of S ∩ N

and constant M > 0. Set

r

:= 1

4 dist K, C\(S ∩ N

)
.

Then, there exist positive constants C,  and m

such that for any z

∈ K and z

∈ S, it holds

|K

(z

, z

)|

e

≤ Cm

e

√m min{r_0,K,|z0−z1|}

where we assume m ≥ max{m

, M − 1} and n ≥ m − M + 1. The constants C, 

and m

only depend on Q, K and M .

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Scientific papers

25

Paper A

27

Antti Haimi and Haakan Hedenmalm

Abstract. For integersn, q = 1, 2, 3, . . ., let Poln,qdenote the C-linear space of polynomials in z and ¯z, of degree ≤ n − 1 in z and of degree ≤ q − 1 in ¯z. We supply Poln,qwith the inner product structure of

L²

C, e^−m|z|2dA

, where dA(z) = π⁻¹dxdy, z = x + iy;

the resulting Hilbert space is denoted by Polm,n,q. Here, it is assumed that m is a positive real. We let Km,n,qdenote the reproducing kernel of Polm,n,q, and study the associated determinantal process, in the limit as m, n → +∞ while n = m + O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble – the eigenvalue process of random (normal) matrices with Gaussian weight. An interpretation is that of permitting a few higher Landau levels. We consider local blow-ups of the polyanalytic Ginibre ensem- bles around points in the spectral droplet, which is here the closed unit disk D¯ := {z ∈ C : |z| ≤ 1}. We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m^−1/2; the typical distance is the same both for interior and for boundary points of ¯D. This amounts to obtaining the asymp- totical behavior of the generating kernel Km,n,q. Following [2], the asymptotics of the Km,n,qis rather conveniently expressed in terms of the Berezin measure (and density)

dB^hzim,n,q(w) := B^hzim,n,q(w)dA(w), B^hzim,n,q(w) =|Km,n,q(z, w)|² Km,n,q(z, z) e^−m|w|2. For interior points |z| < 1, we obtain that dB^hzim,n,q(w) → dδzin the weak-star sense, where δzdenotes the unit point mass at z. Moreover, if we blow up to the scale of m^−1/2around z, we get convergence to a measure which is Gaussian for q = 1, but exhibits more complicated Fresnel zone behavior for q > 1. In contrast, for exterior points |z| > 1, we have instead that dB^hzim,n,q(w) → dω(w, z, D^e), where dω(w, z, D^e) is the harmonic measure at z with respect to the exterior disk D^e:= {w ∈ C : |w| > 1}. For boundary points, |z| = 1, the Berezin measure dB^hzim,n,q

converges to the unit point mass at z, like for interior points, but the blow-up to the scale m^−1/2exhibits quite different behavior at boundary points compared with interior points. We also obtain the asymptotic boundary behavior of the 1-point function at the coarser local scale q^1/2m^−1/2.

Key words and phrases. Bargmann-Fock space, polyanalytic function, determinantal point process.

The second author is supported by the Göran Gustafsson Foundation (KVA) and Vetenskapsrådet (VR).

1. Introduction

Notation. We will write use standard notation, such as ∂X and int(X) for the boundary and the interior of a subset X of the complex plane C. The complex conjugate of a complex number z is usually written as ¯z. We write R for the real line, D := {z ∈ C : |z| < 1} for the open unit disk, and D^e:= {z ∈ C : |z| > 1} for the open exterior (punctured) disk. The characteristic function of a set E is written 1E. We write

dA(z) = π⁻¹dxdy, where z = x + iy ∈ C,

for the normalized area measure in C, and use the standard Wirtinger derivatives

∂z:=¹₂(∂x− i∂y), ¯∂z:=¹₂(∂x+ i∂y), where z = x + iy.

We also write ∆ for the (quarter) Laplacian

∆z:= ∂_z¯∂z=¹₄(∂²_x+∂²_y).

Determinantal projection processes. Given a locally compact topological space X with a Radon measure µ, a determinantal projection process (in the sequel just determinantal process) is a random configuration of n points defined by the following probability measure on Xⁿ:

(1.1) dP(z1, . . . ,zn) = 1

n!det[Kn(zi,zj)]ⁿ_i,j=1dµ(z1) · · · dµ(zn).

Here, Knis the integral kernel of a projection operator to an n-dimensional sub- space of L²(X, µ). It is customary to identify all the permutations of the points and think the process as a random measurePn

j=1δzjon X.

A general definition of a determinantal process was introduced by Macchi [21], who wanted to model fermions in quantum mechanics. Indeed, for any determinantal process, the probability density vanishes whenever any two points in the n-tuple (z1, ...,zn) coincide (fermions are forbidden to be in the same state).

We interpret this as saying that the points in the n-tuple repel each other. Point processes of this kind appear in several contexts, e.g., in random matrix theory and combinatorics (for general surveys, s [19], [9]; we should also mention the books [22], [10], [11], [5], [14]).

Eigenvalues of random normal matrix ensembles. Our main motivating exam- ple comes from the theory of random normal matrices. This topic has in re- cent years been subject to rather active investigation by physicists as well as by mathematicians. For an introduction, see, e.g., [31]. So, we shall use X = C and dµ(z) = e^−mQ(z)dA(z), for a positive weight function Q satisfying some mild regu- larity and growth conditions; m is a positive real parameter, and dA(z) = π⁻¹dxdy is the normalized area measure. Let us write L²(C, e^−mQ) := L²(X, µ) in this situ- ation. The determinantal projection process is associated with an n-dimensional subspace of L²(C, e^−mQ), and we will use the space Polnof all polynomials in z of degree ≤ n − 1; we write Polm,nto indicate that we have supplied Polnwith the Hilbert space structure of L²(C, e^−mQ). The density of the process is then given by the reproducing kernel Km,nof the space Polm,n. So, we are talking about the probability measure

(1.2) dP(z1, . . . ,zn) = 1

n!det[Km,n(zi,zj)]ⁿ_i,j=1e^{−m{Q(z1)+···+Q(zn)}}dA(z1) · · · dA(zn).