# Rescaling Ward Identities in the Random Normal Matrix Model

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## Abstract

We study spacing distribution for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward’s (or the “rescaled loop”) equation—an identity satisfied by all sequential limits of the rescaled one-point functions.

## Keywords

Random normal matrix Universality Ward’s equation Translation invariance## Mathematics Subject Classification

Primary 60B20 Secondary 60G55 81T40 30C40 30D15 35R09*M*(i.e., \(MM^*=M^*M\)) of some large order

*n*, picked randomly with respect to a probability measure of the form

*dM*is the surface measure on normal \(n\times n\) matrices inherited from \(2n^2\)-dimensional Lebesgue measure via the natural embedding into \({{\mathbb {C}}}^{\, n^2}\), \(Q(\zeta )\) is a suitable real-valued function defined on \({{\mathbb {C}}}\) (“large” as \(\zeta \rightarrow \infty \)), and \(\mathcal {Z}_n\) is a normalizing constant; \({\text {tr}}Q(M)=\sum _1^n Q(\zeta _j)\) is the usual trace of the matrix

*Q*(

*M*), where \(\zeta _j\) denote the eigenvalues.

A random sample \(\{\zeta _j\}_1^n\), where \(\zeta _j\) are the eigenvalues of *M*, is known as a *random normal eigenvalue ensemble*. It is convenient to briefly call it a “system.”

If *Q* is just defined on \({{\mathbb {R}}}\) and *dM* is surface measure on the Hermitian matrices (i.e., \(M^*=M\)), we have random Hermitian matrices. The study of such ensembles, e.g., using the technique of Riemann–Hilbert problems, is a classical and active area of research.

As \(n\rightarrow \infty \), the system tends to occupy a certain compact set *S* called the *droplet*. We will here fix a point \(p\in S\) and study microscopic properties of the system near *p*. This corresponds to the study of spacing distribution in Hermitian theory.

The simplest case is the well-known Ginibre ensemble, in which \(Q(\zeta )=\left| {\,\zeta \,} \right| ^{\,2}\) and *S* is the closed unit disc. In this case, the system \(\{\zeta _j\}_1^n\) can be interpreted as the eigenvalues of an \(n\times n\) matrix whose entries are i.i.d. complex, centered Gaussian random variables of variance 1 / *n*. We note a few facts about this ensemble.

*S*, by letting \(z_j = \sqrt{n} \zeta _j,\) and let \(n\rightarrow \infty \), then simple calculation shows that the rescaled system \(\{z_j\}_1^n\) converges to a determinantal random point field with the correlation kernel

*G*as the

*Ginibre kernel*and the corresponding point field as the “bulk Ginibre point field.”

*F*is the entire function given by

*F*is closely related to the “plasma dispersion function” in the physics literature, see [19]. Sometimes we will refer to

*F*as the

*plasma function*(Figs. 1, 2).

*hard edge*Ginibre ensemble, where the potential is redefined as \(+\infty \) outside the unit disc

*S*. In this case, we get

*left*half-plane \({{\mathbb {L}}}=\{\,z\,;\, {\text {Re}}z<0\,\}\),

*F*is given by (0.3) (cf. Fig. 2).

*p*in the interior of

*S*with \(\Delta Q(p) > 0,\) then the rescaled processes converge to the bulk Ginibre point field.

Full universality at boundary points remains an open problem, but we will here obtain some partial results by suggesting and exploring a new approach that is based on rescaling of Ward’s identities (or loop equations). This approach is of interest also in other contexts, see, e.g., [1, 8].

*Q*) equation that is satisfied by all sequential limits

*K*of the rescaled correlation kernels. In the case of a regular point on the free boundary, the equation has the following form:

*Berezin kernel*

*B*(

*z*,

*w*) (see Fig. 3),

*R*determines

*K*and therefore

*B*and

*C*by means of analytic continuation.

The analysis of equation (0.6) is one of the main points of this paper. We will describe all (vertically) *translation invariant* solutions *R* to Ward’s equation, i.e., solutions for which *R*(*z*) depends only on \({\text {Re}}z\). If we rescale properly about a regular boundary point, the boundary of the droplet will approach the imaginary axis, making it very plausible that the intensity of eigenvalues should be translation invariant. We will obtain results that strongly indicate that this is in fact the case, but we are not able to completely settle the issue here. It is however easy to verify translation invariance for radially symmetric potentials, so we do get universality for that class.

The method of rescaled Ward identities is quite general and can be used in a variety of different settings. In addition to the already mentioned cases of regular points in the bulk, or on a free or hard edge boundary, one can consider various types of singular points as well. We get certain equations that depend only on the setting but not on *Q* (“universality”). We will derive such Ward’s equations in several settings but will focus most on the regular free boundary case (0.2). The analysis of other cases appear elsewhere, see [5, 6, 8].

A detailed description of our results is given in the following section.

**Notational conventions**. By \(D(\zeta ;r)\) we denote the open disc with center \(\zeta \) and radius *r*. We write \({\partial }E\), \({\text {Int}}E\), \({\text {cl}}E\), and \( E^c\) for the boundary, the interior, the closure, and the complement of a set \( E\subset {{\mathbb {C}}}\). The indicator function of a set *E* is denoted by \({\mathbf {1}}_E\). We write \({\partial }=\frac{1}{2}\left( {\partial }/{\partial }x-i{\partial }/{\partial }y\right) \) and \({\bar{\partial }}=\frac{1}{2}\left( {\partial }/{\partial }x+i{\partial }/{\partial }y\right) \) for the complex derivatives and \(\Delta ={\partial }{\bar{\partial }}\) for the *normalized* Laplacian. Thus \(\Delta \) is \(\frac{1}{4}\) times the standard Laplacian. We write \(dA(z)=d^{\,2} z/\pi \) for normalized Lebesgue measure. Thus the unit disc has measure 1. The volume measure on \({{\mathbb {C}}}^k\) is defined by \(dV_k(\zeta _1,\ldots ,\zeta _k)=dA(\zeta _1)\cdots dA(\zeta _k)\). The intensities and the correlation kernel of the original point processes are boldfaced as \({\mathbf R}_{n,k}\) and \({\mathbf {K}}_n\), respectively, and those of the rescaled point processes are italicized as \(R_{n,k}\) and \(K_n\), respectively.

A continuous function \(f:{{\mathbb {C}}}^2\rightarrow {{\mathbb {C}}}\) is termed *Hermitian* if \(f(z,w)=\overline{f(w,z)}\). We shall say that *f* is *Hermitian-analytic* (or *Hermitian-entire*) if *f* is Hermitian and analytic (entire, respectively) as a function of *z* and \(\bar{w}\). A Hermitian-entire function is uniquely determined by its values *f*(*z*, *z*) on the diagonal, see [23, Lemma 2.5.1]. A Hermitian function *c* is called a *cocycle* if \(c(z,w)=g(z)\overline{g(w)}\) for a continuous unimodular function *g*. The determinant of a matrix \((m_{ij})\) remains unchanged if we multiply each \(m_{ij}\) by \(c(z_i,z_j)\).

## 1 Introduction and Results

### 1.1 Potential Theory and Droplets

Fix a suitable function (“*external potential*”) \(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\cup \{+\infty \}\). Let \({\mathcal P}\) denote the class of positive, compactly supported Borel measures on \({{\mathbb {C}}}\).

*weighted logarithmic energy*of \(\mu \in {\mathcal P}\) in external field

*Q*by

*Q*.

We always assume that *Q* is *lower semi-continuous* and that *Q* is finite on some set of positive logarithmic capacity. We will also assume that *Q* is sufficiently large at infinity so as to confine the system to a finite portion of the plane. To be precise, it suffices to assume that \(\liminf _{\zeta \rightarrow \infty }Q\left( \zeta \right) /\log \left| {\,\zeta \,} \right| >2.\)

*equilibrium measure*\(\sigma \) that minimizes the weighted energy,

*S*as the

*droplet*in the external field

*Q*. It is known that if

*Q*is smooth in some neighborhood of

*S*, then \(\sigma \) is absolutely continuous and takes the explicit form (see [32])

*S*.

*throughout*are that there is some neighborhood \(\Omega \) of

*S*such that

*regular*in the sense that there is a disc \(D=D(p;\epsilon )\) such that \(D\setminus S\) is a Jordan domain and \(D\cap ({\partial }S)\) is a real analytic arc. A nonregular point \(p\in {\partial }S\) is called a

*singular*boundary point. Such points can be classified further as cusps or double points.

### 1.2 Rescaling Eigenvalue Ensembles

*Boltzmann–Gibbs law*,

We can thus think of the point-process \((\zeta _j)_1^n\) either as a system of repelling point-charges in external field *nQ*, with logarithmic interactions, or as eigenvalues of random normal matrices from the ensemble above.

*determinantal*. This means that if \({\mathbf R}_{n,k}\) denotes the

*k*-point intensity function of the process \(\{\zeta _j\}_1^n\), then

*correlation kernel*. Indeed, by Dyson’s determinant formula, given in, e.g., [32, Section IV.7.2] or [17, 28], we have the formula

*j*:th orthonormal polynomial with respect to the measure \(e^{-nQ(\zeta )}dA(\zeta )\).

*k*-point function of \({{\mathbb {P}}}_n\) is defined for distinct points \(\eta _j\) by

*B*; \({{\mathbb {E}}}_n\) is expectation with respect to \({{\mathbb {P}}}_n\).

It should be mentioned that the so-called \(\beta \)-*ensembles* corresponding to the Boltzmann–Gibbs law \(d{{\mathbb {P}}}_n^{(\beta )}(\zeta ) \propto e^{\,-\beta \,H_n(\zeta )}\,d V_n(\zeta )\) are not determinantal if \(\beta \ne 1\), and some aspects of the methods we develop in this manuscript do not work for them.

We are interested in microscopic properties of the system \(\{\zeta _j\}_1^n\) near a point \(p\in S\). It is natural to magnify distances about *p* by a factor \(\sqrt{n\Delta Q(p)}\). We also fix an angle \(\theta \in {{\mathbb {R}}}\); when *p* is a regular boundary point, we choose \(\theta \) so that \(e^{i\theta }\) as the *outer normal* to \({\partial }S\) at *p*; in other cases, we may choose it arbitrarily.

*n*-dependent point \(p = p_n,\) or alternatively, let 0 denote the origin in an

*n*-dependent coordinate system. This generalization presents no new difficulties as long as the sequence \(p_n\) is contained in a sufficiently small neighborhood of the droplet and satisfies the decisive condition \(\Delta Q(p_n) \ge \text {const.} > 0.\) See [1, 6].

*law*of \(\Theta _n\) is defined as the image of the Boltzmann–Gibbs measure (1.1) under the map (1.4). The rescaled system \(\Theta _{n}\) then has intensities denoted by \(R_{n,k}\), where

*k*-point function for a “point field” in \({{\mathbb {C}}}\), meaning a probability law on a suitable space of infinite configurations \(\{z_i\}_1^\infty \subset {{\mathbb {C}}}\) (cf. [34], see also a forthcoming version of [6]).

*K*locally uniformly on \({{\mathbb {C}}}^2\). Moreover, the limiting point field is uniquely determined by

*K*if the functions \(K_n(z,z)\) are uniformly bounded, and it is then determinantal with intensity functions

*K*with bounded convergence on the diagonal in \({{\mathbb {C}}}^2\).

### 1.3 Compactness, Nontriviality, and Ward’s Equation

Our first theorem states the existence of sequential limits of the rescaled point processes \(\Theta _n\) and specifies the form of limiting correlation kernels.

## Theorem 1.1

- (i)
There is a sequence \(c_n\) of cocycles such that every subsequence of \(\left( c_nK_n\right) _1^\infty \) has a further subsequence that converges uniformly to some limit

*K*on compact subsets of \({{\mathbb {C}}}^2.\) - (ii)
Each

*K*in (i) has the form \(K=G\Psi \), where \(\Psi \) is a Hermitian entire function and*G*is the Ginibre kernel.

*limiting kernel*

*K*in Theorem 1.1 is a positive kernel in Aronszajn’s sense [9]; i.e., for all finite sequences \((z_j)_1^N\) of points and all choices of scalars \((\alpha _j)_1^N\), we have

## Theorem 1.2

- (i)
*K*satisfies the following “mass-one inequality”:$$\begin{aligned} \text {for all } z \in {{\mathbb {C}}}, \qquad \int _{{\mathbb {C}}}|K(z,w)|^2\,dA(w) \le K(z,z). \end{aligned}$$(1.6) - (ii)
Let \(L(z,w):=e^{z{\bar{w}}} \Psi (z,w).\) Then

*L*is the reproducing kernel for a certain Hilbert space \({\mathcal H}_*\) of entire functions that sits contractively in \(L^2_a(\mu )\). - (iii)Also we havein the sense that$$\begin{aligned} 0\le K \le G \end{aligned}$$
*K*and \(G-K\) are positive kernels.

*K*is the correlation kernel of some point field in the plane, which we call a

*limiting point field*. The 1-point function of this point field is denoted by \(R(z)=K(z,z)\). If

*R*does not vanish on \({{\mathbb {C}}},\) we can consider the Berezin kernel \(B(z,w) = {|K(z,w)|^2}/{K(z,z)}\) and define the Cauchy transform

## Theorem 1.3

- (i)
Either

*R*is trivial, in the sense that \(R=0\) identically in \({{\mathbb {C}}}\), or \(R>0\) everywhere. - (ii)Ward’s equation: if
*R*is nontrivial, then the integral*C*(*z*) in (1.7) converges and defines a smooth function such that$$\begin{aligned} {\bar{\partial }}C(z)=R(z)-1-\Delta \log R(z),\qquad z\in {{\mathbb {C}}}. \end{aligned}$$

The first part of this theorem says that we have a dichotomy (some sort of “zero-one law”). Both cases are possible. The next theorem implies that we have nontriviality at regular boundary points, whereas triviality (i.e. \(R\equiv 0\)) occurs, for example, if *p* is a singular boundary point, see [6].

### 1.4 A Priori Estimates

Ward’s equation has many solutions. In order to fix a solution uniquely, we need to know that certain additional conditions are satisfied. These conditions depend on the nature of the point we are rescaling about. For regular points, the results are as follows.

## Theorem 1.4

*K*be a corresponding limiting kernel in Theorem 1.1. Write \(x={\text {Re}}z\).

- (i)Exterior estimate: There is a constant
*C*such that$$\begin{aligned} R(z)\le C\,e^{\,-\,2\,x^{\,2}}\quad \text {for}\quad x\ge 0. \end{aligned}$$ - (ii)Interior estimate: If \(\ell \) is any number with \(\ell <1/2\), then there is a constant
*C*(depending on \(\ell \)) such that$$\begin{aligned} 0 \le 1- R(z) \le C\,e^{\,-\,\ell \, x^{\,2}}\quad \text {for}\quad x\le 0. \end{aligned}$$

## Theorem 1.5

*S*is connected and all boundary points of

*S*are regular. For almost all boundary points, the following identity holds for all limiting kernels:

### 1.5 Translation Invariant Solutions to Ward’s Equation

*K*) is (vertically)

*translation invariant*(in short:

*t.i.*) if

*f*be a bounded function on \({{\mathbb {R}}}\). Then we define the entire function \(\gamma *f\) by the equation

*t.i.*, the entire function \(\Phi \) in (1.8) in has the form \(\Phi = \gamma *f\) with a bounded function

*f*.

## Theorem 1.6

Let \(\Phi =\gamma *f\), where *f* is a bounded function. Then the kernel \(K(z,w)=G(z,w)\Phi (z+\bar{w})\) satisfies Ward’s equation if and only if \(\Phi =\gamma *{\mathbf {1}}_I\) for some connected set \(I\subset {{\mathbb {R}}}.\)

Using a priori estimates in Theorem 1.4, we obtain the following corollary.

## Theorem 1.7

*K*at a regular boundary point is translation invariant, then \(K(z,w)=G(z,w)\Phi (z+\bar{w})\) and there is some

*a*such that

Since \(a=0\) is the only choice of *a* consistent with the 1 / 8-formula in Theorem 1.5, we have the following result for radially symmetric potentials, \(Q(z)=Q(|z|)\).

## Theorem 1.8

Assume that the droplet *S* is connected. If *Q* is radially symmetric and *p* is any boundary point, then the rescaled point processes converge to the boundary Ginibre point field.

As we mentioned, it is natural to expect that the boundary Ginibre point process is universal at all regular boundary points for all potentials. A possible approach to ruling out the existence of non-translation-invariant solutions, which satisfy the given apriori estimates, is outlined in Section 8.3.

Recently (after this work was completed) Lee and Riser [25] established this type of convergence for the nonsymmetric “elliptic” potentials \(Q=|\,\zeta \,|^{\, 2}-t{\text {Re}}(\zeta ^{\,2})\), \(0<t<1\). Their proof depends on explicit computations with the orthogonal polynomials in [31]. Very recently, it seems that universality has been verified in a rather satisfactory generality, by Hedenmalm and Wennman [21]. The approach there depends on a new asymptotic formula for planar orthogonal polynomials. Our approach is quite different, and in fact both methods seem to be of independent interest.

### 1.6 Berezin Kernel and Mass-One Equation

*z*,

*n*-point process and \(R_{n-1}^{\,(z)}\) is the 1-point function for the conditional \((n-1)\)-point process \(\Theta _n|\left\{ z\in \Theta _n\right\} \).

*K*is a limiting kernel, we define

*K*satisfies the

*mass-one equation*if for all

*z*,

We can describe solutions to the mass-one equation in the *t.i. * case.

## Theorem 1.9

Let \(\Phi =\gamma *f\), where *f* is a bounded function. Then the kernel \(K(z,w)=G(z,w)\Phi (z+\bar{w})\) satisfies the mass-one equation if and only if there is a Borel set \(E \subset {{\mathbb {R}}}\) of positive measure such that \(f={\mathbf {1}}_E\).

### 1.7 Organization of the Paper

In Section 2, we consider the boundary Ginibre ensembles, both for the free boundary and the hard edge. We give a short proof of the convergence of rescaled ensembles to the boundary Ginibre point fields with kernels (0.2) and (0.4), respectively.

In Section 3, we prove Theorem 1.1 (compactness and analyticity) and Theorem 1.2 (mass-one inequality and positivity of *K* and \(G-K\)).

In Section 4, we derive Ward’s equation and prove Theorem 1.3.

In Section 5, we establish a priori bounds for regular points (Theorem 1.4). We also prove the 1 / 8-formula in Theorem 1.5 at almost every boundary point.

In Section 6, we specialize to *t.i. * solutions and prove Theorems 1.6–1.9.

In Section 7, we write down versions of Ward’s equation in some settings (hard edge, bulk singularities, and \(\beta \)-ensembles) that are different from the free boundary case discussed above.

The last section, Section 8, contains some general remarks. We relate our methods and results to classical asymptotics for sections of power series. We discuss Hilbert spaces of entire functions associated with limiting kernels. We also comment on the nature of the mass-one equation and Ward’s equation and show that they take the form of twisted convolution equations.

## 2 The Ginibre Ensembles

### 2.1 Principles of Notation

Consider first a general potential *Q*. By a *weighted polynomial of order* *n*, we mean a function of the form \(f=q\cdot e^{-nQ/2}\), where *q* is an (analytic) polynomial of degree at most \(n-1\). Let \(\mathcal {W}_n\) denote the space of all weighted polynomials of order *n*, considered as a subspace of \(L^2=L^2({{\mathbb {C}}},dA)\). It is well known that the reproducing kernel \({\mathbf {K}}_n(\zeta ,\eta )\) for the space \(\mathcal {W}_n\) is a correlation kernel for the process \(\{\zeta _j\}_1^n\) corresponding to *Q*. This implies that one has the formula (1.3) for \(\mathbf {K}_n(\zeta ,\eta ).\) Recall the Ginibre potential \(Q=\left| \,\zeta \,\right| ^{\,2}\). The corresponding droplet is \(S=\left\{ \,\zeta \,;\,\left| \,\zeta \,\right| \le 1\,\right\} \).

We shall give an elementary proof for convergence to the boundary Ginibre point field using Poisson approximation of the normal distribution. Our proof is somewhat similar to the argument in the paper [30], where the spectral radius of a Ginibre matrix is studied.

### 2.2 Free Boundary Ginibre Ensemble

Let \(\{\zeta _j\}_1^n\) denote a random configuration for the free boundary Ginibre process. We rescale about the boundary point \(p=1\) in the outer normal direction, via \(z_j=\sqrt{n}\left( \zeta _j-1\right) \), writing \(\Theta _n=\{z_j\}_1^n\) for the rescaled process. We shall prove the following theorem, found in [18] (cf. [12]).

## Theorem 2.1

The processes \(\Theta _n\) converge to the boundary Ginibre point field as \(n\rightarrow \infty \) with locally uniform convergence of intensity functions.

Since \(K(z,z)<1\), it suffices to prove the statement about convergence of intensity functions.

*i.e.*,

## Lemma 2.2

*G*is the Ginibre kernel and \(o(1)\rightarrow 0\) uniformly on compact sets as \(n\rightarrow \infty \).

## Proof

*K*is the free boundary kernel defined in (0.2). Since the factor \(c_n(z,w)=e^{i\sqrt{n}{\text {Im}}(z-w)}\) is a cocycle, this factor can be dropped when computing intensity functions \(R_{n,k}(z)=\det (K_n(z_i,z_j))\). This proves the desired convergence of intensity functions, at the same time establishing existence and uniqueness of the boundary Ginibre point field. The proof of Theorem 2.1 is complete. \(\square \)

### 2.3 Hard Edge Ginibre Ensemble

Let \(Q^S(z)=\left| {\,z\,} \right| ^{\,2}\) when \(|\,z\,|\le 1\) and \(Q^S=+\infty \) otherwise. Let \(\{\zeta _i\}_1^n\) denote a random configuration from the corresponding ensemble. Rescaling about \(p=1\) via \(z_j=\sqrt{n}\,(\zeta _j-1)\), we obtain a process \(\Theta _n\). Let *H* be the hard edge function (0.4).

## Theorem 2.3

Note that \(K(z,z)<2\) for all *z*. As we mentioned in Subsection 1.2, the convergence of intensity functions in the theorem implies the existence and uniqueness of a field \(BG_h\) with correlation kernel *K*. It thus suffices to prove convergence.

*j*. We have shown that

## 3 Analyticity and Compactness

In this section, we prove Theorems 1.1 and 1.2. We start by introducing appropriate notation; after that we will deduce our results using normal families coupled with some theory for reproducing kernels.

### 3.1 General Notation

We denote by *C* (“large”) and *c* (“small”) various positive unspecified constants (independent of *n*) whose exact value can change meaning from time to time.

### 3.2 Potentials and Reproducing Kernels

Fix a neighborhood \(\Omega \) of \(S\) and a number \(\delta _0>0\) such that *Q* is real-analytic and strictly subharmonic in the \(2\delta _0\)-neighborhood of \(\Omega \).

### 3.3 Bulk Approximations

### 3.4 Auxiliary Estimates

We recall first the following simple pointwise-\(L^2\) estimate.

## Lemma 3.1

*u*is analytic in the disc \(D:=D\left( p;c/\varrho _n\right) \), (\(\varrho _n=\sqrt{n\Delta Q(p)}\)), where

*Q*is \(C^2\)-smooth at

*p*. Let \(f=ue^{-\,nQ/2}\). Then there is a number

*C*depending only on

*c*and \(\Delta Q(p)\) such that

## Proof

Fix a number \(a>1\) and consider the function \(F_n(z)=f\left( p+z/\varrho _n\right) \cdot e^{a\left| {\,z\,} \right| ^{\,2}/2}\). We have \(\Delta \log \left| {\, F_n(z)\,} \right| ^{\,2}\ge -\Delta Q\left( p+z/\varrho _n\right) /\Delta Q\left( p\right) +a> 0\) for \(\left| {\,z\,} \right| \le c\) if *n* is large enough. Hence \(|F_n|^2\) is subharmonic in *D*, which implies the desired estimate. \(\square \)

*obstacle function*corresponding to

*Q*, we mean the subharmonic function

*S*while \(\check{Q}\) is harmonic on \(S^c\) and is of logarithmic increase

## Lemma 3.2

If \(f\in {\mathcal W}_n\) and \(\left| {\,f\,} \right| \le 1\) on *S*, then \(\left| {\, f\,} \right| \le e^{\,-\,n\left( Q-\check{Q}\right) /2}\) on \({{\mathbb {C}}}\).

## Proof

If \(f=p\cdot e^{-nQ/2}\), then \(\frac{1}{n} \log \left| {\, p\,} \right| ^{\,2}\) is a subharmonic minorant of *Q* that grows no faster than \(\log \left| {\,\zeta \,} \right| ^{\,2}+{\text {const.}}\) as \(\zeta \rightarrow \infty \). It is well known that \(\check{Q}(\zeta )\) is the supremum of \(f(\zeta )\) where *f* ranges over the functions having these properties (see, e.g., [32]). \(\square \)

## Lemma 3.3

There a constant \(C=C[Q]\) such that \({\mathbf R}_n(\zeta )\le Cne^{\,-\,n\left( Q-\check{Q}\right) (\zeta )}\).

### 3.5 Convergence of Approximate Kernels

*Q*. A kernel \(K_n\) for the rescaled process \(\{z_j\}_1^n\) (about \(p=0\) in positive real direction) is given by

*z*,

*w*) such that \((\zeta ,\eta )\in \Lambda \). Note that there is a constant \(\rho >0\) depending only on \(\Delta Q(0)\) such that

## Lemma 3.4

We have \(K_n^\#(z,w)=c_n(z,w)G(z,w)(1+o(1))\) as \(n\rightarrow \infty \), where \(c_n\) are cocycles on \(V_n\) and \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \), uniformly on compact subsets of \({{\mathbb {C}}}^2\).

## Proof

## Remark

### 3.6 Compactness

## Lemma 3.5

The function \(\Psi _n\) is Hermitian-analytic in the set \(V_n\).

## Proof

*z*and

*w*. \(\square \)

## Lemma 3.6

For each compact set \(K\subset {{\mathbb {C}}}^2\), there is a constant \(C=C_K\) such that \(\left| \,\Psi _n(z,w)\,\right| ^{\,2}\le Ce^{\,|z-w|^2}\) when \((z,w)\in K\) and *n* is large enough.

## Proof

*K*,

## Proof of Theorem 1.1

*z*and recall that \(\int \left| {\,K_{n_k}(z,w)\,} \right| ^{\,2}\, dA(w)=K_{n_k}(z,z)\). In terms of the functions \(\Psi _n\),

Finally we use Lemma 3.4 to select cocycles \(c_n\) such that \(c_nK_n^\#\rightarrow G\) uniformly on compact subsets of \({{\mathbb {C}}}^2\). Then \(c_{n_k}K_{n_k}=\Psi _{n_k}\cdot c_{n_k}K_{n_k}^\#\rightarrow G\Psi =K\), finishing the proof of the theorem. \(\square \)

## Lemma 3.7

## Proof

## Remark

### 3.7 Holomorphic Kernels and Positivity

In this subsection, we prove Theorem 1.2. Part (i), the mass-one inequality, is already explained in the proof of Theorem 1.1.

*L*is the reproducing kernel for a certain Hilbert space \({\mathcal H}_*\) of entire functions that is contractively embedded in the Fock space \(L^2_a(\mu )\) of entire functions square-integrable with respect to the measure

*complementary*holomorphic kernel \(\tilde{L}(z,w):=e^{z\bar{w}}(1-\Psi (z,w))\) is a positive kernel. Our proof of the latter fact depends on a theorem of Aronszajn on differences of reproducing kernels.

*f*to a smooth function on \({{\mathbb {C}}}^2\) in some way. We will require that the extended function satisfies \(f(\zeta ,\zeta )=Q(\zeta )-\left| {\zeta } \right| ^2-2{\text {Re}}H(\zeta )\) for all \(\zeta \in {{\mathbb {C}}}\). (The function

*H*is of course well-defined everywhere, being a second-degree polynomial.) It is important to observe that \(f(\zeta ,\zeta )=O\left( |\,\zeta \,|^{\,3}\right) \) as \(\zeta \rightarrow 0\).

*n*-dimensional Hilbert space

## Lemma 3.8

\( L_n\) is the reproducing kernel for \({\mathcal H}_n\) and \(L_n\rightarrow L\) locally uniformly on \({{\mathbb {C}}}^2\) as \(n\rightarrow \infty \). Moreover, for all \(z\in {{\mathbb {C}}}\), we have \(\int _{{\mathbb {C}}}\left| {L(z,w)} \right| ^2e^{-|w|^2}\, dA(w)<\infty .\)

## Proof

*O*-constant is uniform on compact subsets of \({{\mathbb {C}}}^2\). Moreover, the mass-one inequality shows that

We can now finish the proof of part (ii) of Theorem 1.2.

*z*, so the inner product is actually a true (positive definite) inner product. By Fatou’s lemma and the convergence \(L_n\rightarrow L\), we now derive a basic inequality (where \(d\mu (z)=e^{-\,|\,z\,|^{\,2}}\,dA(z)\)):

It follows that the completion \({\mathcal H}_*\) of \({\mathcal M}\) can be regarded as a (possibly non-closed) subspace of \(L^2_a(\mu )\). We will write \({\mathcal H}_\Psi \) for \({\mathcal H}_*\) and speak of the space of entire functions associated with the kernel \(L(z,w)=e^{\,z\bar{w}\,}\,\Psi (z,w)\). (The mass-one equation is equivalent to that the inclusion *I* be isometric; see Subsection 8.2.)

## 4 Ward’s Equation and the Mass-One Inequality

In this section, we prove part (ii) of Theorem 1.3. We start by deriving a slightly modified (or “localized,”) form of the Ward identity used in [3]. This modification is necessary when dealing with hard edge processes and is in general quite convenient.

The main theme of the section is the rescaled form of this identity and the passage to subsequential limits. In the process of proving this convergence, we will for instance verify the zero one-law for limiting one-point functions.

### 4.1 Ward’s Identity

*n*variables by

*Q*is smooth in a neighborhood of the support of \(\psi \). We can then make sense of \(W_n^+[\psi ]\) even though \({\partial }Q\) may be undefined in portions of the plane. Indeed, we

*define*

*Ward’s identity*.

## Theorem 4.1

\({{\mathbb {E}}}_n\, W_n^+[\psi ]=0.\)

## Proof

We modify the argument in [3]. Given \(\zeta \in {{\mathbb {C}}}\) and \(\varepsilon >0\), we let \(D_{\varepsilon \psi }(\zeta )\) be the closed disc centered at \(\zeta \) of radius \(\varepsilon \left| {\psi (\zeta )} \right| \). Choosing \(\varepsilon =\varepsilon (\psi )>0\) sufficiently small, there are two alternatives for each point \(\zeta \in {{\mathbb {C}}}\): (i) *Q* is \(C^2\)-smooth in a neighborhood \(D_{\varepsilon \psi }(\zeta )\) of \(\zeta \), or (ii) \(\psi (\zeta )=0\).

*Q*is \(C^2\)-smooth, then, by Taylor’s formula,

*j*’s we have \(\psi (\zeta _j)=0\) and \(\eta _j=\zeta _j\), whence (4.2) holds, since \(\left[ {\partial }Q\cdot \psi \right] (\zeta _j)=0\) by definition. Hence (4.2) holds in all cases, so

### 4.2 Rescaled Version

*p*in a small neighborhood \(\Omega \) of

*S*, where

*Q*is \(C^2\)-smooth. We rescale the system \(\left\{ \zeta _j\right\} _1^n\) about

*p*in the usual way, obtaining the rescaled system \(\Theta _n=\{z_j\}_1^n\), where \(z_j=e^{\,-i\theta }\sqrt{n\Delta Q(p)}\,(\zeta _j-p).\) The Berezin kernel rooted at

*p*is defined by

## Theorem 4.2

## Proof

### 4.3 Ward’s Equation

*R*does not vanish identically.

*holomorphic kernel*

## Lemma 4.3

*L*(*z*, *w*) is a positive kernel, and \(z\mapsto L(z,z)\) is logarithmically subharmonic; i.e., the function \(|z|^2+\log R(z)\) is subharmonic.

## Proof

Since \(K(z,w)=L(z,w)e^{-|z|^2/2-|w|^2/2}\) is a positive matrix, i.e., \(\sum \alpha _j\bar{\alpha }_k K(z_j,z_k)\ge 0\) for all choices of points \(z_j\) and scalars \(\alpha _j\), we infer that \(\sum \beta _j\bar{\beta }_k L(z_j,z_k)\ge 0\), where \(\beta _j=\alpha _j e^{-|z_j|^2/2}\); i.e., *L* is a positive kernel. Following Aronszajn [9], we can then define a semi-definite inner product on the span of the \(L_z\)’s by \(\left\langle L_z,L_w\right\rangle _*=L(w,z)\). The completion of the span the functions \(L_z\) (\(z\in {{\mathbb {C}}}\)) is a semi-normed Hilbert space \({\mathcal H}_*\) of entire functions, and the reproducing kernel in this space is *L*.

*L*is Hermitian-entire, the function \(F(z,w)=\left\langle L_w,L_z\right\rangle _*\) is also Hermitian-entire. Moreover, since \({\partial }_z F(z,w)=\left\langle L_w,{\bar{\partial }}_z L_z\right\rangle _*\), \({\bar{\partial }}_w{\partial }_z F(z,w)=\left\langle {\bar{\partial }}_w L_w,{\bar{\partial }}_z L_z\right\rangle _*\), etc., it follows that at points where \(L(z,z)>0\), we have

## Lemma 4.4

*R*does not vanish identically, then all zeros of

*R*are isolated.

## Proof

*j*. However, \({\partial }^j R(z_0)={\partial }_1^j \Psi (z_0,z_0)\) and \({\bar{\partial }}^j R(z_0)={\bar{\partial }}_2^j\Psi (z_0,z_0)\), so the Hermitian function \(\Psi (z,w)\) vanishes whenever \((z-z_0)(w-z_0)=0\). Hence we can write \(\Psi (z,w)=(z-z_0)(\bar{w}-\bar{z}_0)\Psi _1(z,w)\), where \(\Psi _1\) is another Hermitian-entire function. If we define \(\tilde{R}(z)=\Psi _1(z,z)\), we now have \(R(z)=\left| {\,z-z_0\,} \right| ^{\,2}\tilde{R}(z)\).

To prove the second statement, assume that the zeros of *R* have an accumulation point, i.e., that there exists a convergent sequence \((z_j)_1^\infty \) of distinct zeros of *R*. Fix a point *w*, and put \(\psi _w(z)=\Psi (z,w)\). By the argument above, we have that \(\Psi (z,w)=0\) if \(z=z_j\), so the holomorphic function \(\psi _w\) vanishes at all points \(z_j\), whence \(\psi _w\) vanishes identically. Since *w* was arbitrary, \(\Psi =0\), and hence \(R(z)\equiv \Psi (z,z)\equiv 0\). \(\square \)

Note that Lemma 4.3 says that the distribution \(1+\Delta \log R\) is a positive measure.

## Lemma 4.5

In the situation of Lemma 4.4, the measure \(1+\Delta \log \tilde{R}\) is positive in a neighborhood of \(z_0\).

## Proof

*R*, and let \(\chi ={\mathbf {1}}_{D}\) be the characteristic function of some small disc \(D=D(z_0;\varepsilon )\) about \(z_0\). Put \(\mu =\chi \cdot (1+\Delta \log R)\) so \(\mu \) is a positive measure by the previous lemma. By Lemma 4.4, we can write \(\mu =\delta _{z_0}+\chi \cdot (1+\Delta \log \tilde{R})\), so the function

*S*extends analytically to \(z_0\) and is hence subharmonic in

*D*. Otherwise \(S(z_0)=-\infty \). Then

*S*has the sub-mean value property in

*D*. Since

*S*is also upper semicontinuous,

*S*is subharmonic in the entire disc

*D*as desired. \(\square \)

*Z*for the set of isolated zeros of

*R*. When \(z\not \in Z\), we can write \(B(z,w)={\left| {\,K(z,w)\,} \right| ^{\,2}}/{K(z,z)}\) and define

*Z*.

## Lemma 4.6

\(C_{n_k}\) converges boundedly and locally uniformly on \(Z^c\) to *C* as \(k\rightarrow \infty \). In particular, *C* is uniformly bounded on \(Z^c\).

## Proof

*N*such that if \(k>N\), \(|\,z\,|<1/\varepsilon \), \({\text {dist}}(z,Z)\ge \varepsilon \), and \(|\,w\,|<2/\varepsilon \), then

*z*with \({\text {dist}}(z,Z)\ge \varepsilon \), \(|\,z\,|<1/\varepsilon \), it follows that

*Z*. \(\square \)

## Lemma 4.7

*K*is nontrivial. Then Ward’s equation

## Proof

By Theorem 4.2, we know that \({\bar{\partial }}C_{n}=R_n-1-\Delta \log R_n+o(1)\), where “*o*(1)” is some function that converges to 0 uniformly on compacts as \(n\rightarrow \infty \). By Lemma 4.6, the functions \(C_{n_k}\) converge to *C* boundedly and locally uniformly on \(Z^c\). Since *Z* is discrete, this implies that \(\int C_{n_k}f\, dA\rightarrow \int Cf\, dA\) for each test function *f*, viz. \(C_{n_k}\rightarrow C\) in the sense of distributions, and also \({\bar{\partial }}C_{n_k}\rightarrow {\bar{\partial }}C\) in that sense. It follows that the functions \(\Delta \log R_{n_k}\) converge in the sense of distributions. Since \(R_{n_k}\rightarrow R\) locally uniformly, the limit must be \(\Delta \log R\). \(\square \)

## Theorem 4.8

If *R* does not vanish identically, then \(R>0\) everywhere. Moreover, Ward’s equation (4.5) holds pointwise on \({{\mathbb {C}}}\).

## Proof

*v*is smooth in some neighborhood of \(z_0\). If \(C^\mu (z)\) remains bounded as \(z\rightarrow z_0\), then \(\mu =\nu +\delta _{z_0}\) cannot contain any point mass at \(z_0\), so \(\nu \) can be written \(-\delta _{z_0}+\rho \), where \(\rho (\{z_0\})=0\). This contradicts the fact that \(\nu \ge 0\). Hence

*C*in Lemma 4.6. Hence \(R(z_0)= 0\) is impossible.

We have shown that \(R>0\) everywhere. Since Ward’s equation \({\bar{\partial }}C=R-1-\Delta \log R\) holds in the sense of distributions and the right-hand side is smooth, application of Weyl’s lemma now shows that *C*(*z*) is smooth and that Ward’s equation holds pointwise. \(\square \)

### 4.4 Reformulation of Ward’s Equation

## Lemma 4.9

*P*(

*z*) such that

## Proof

*E*such that

### 4.5 Relations for the Boundary Kernel

We finish this section by noting the following theorem.

## Theorem 4.10

The kernel \(K(z,w)=G(z,w)F(z+\bar{w})\) satisfies Ward’s equation and the mass-one equation.

## Proof 1

The proof of Ward’s equation in Subsection 4.3 and the example of the Ginibre ensemble in Subsection 2.2 show that Ward’s equation is satisfied. The mass-one equation can be deduced in a similar way; in fact, we shall prove in Subsection 6 that the mass-one equation is a consequence of Ward’s equation in the translation invariant case. \(\square \)

## Proof 2

*equivalent*equation

## 5 A Priori Estimates at Regular Boundary Points

In this section, we prove the a priori estimates for the one-point function in Theorem 1.4. We will develop the technique of “approximate Bergman projections,” which has the advantage of being relatively simple while still giving good enough a priori estimates of the main term in the one-point function \({\mathbf R}_n\). (A similar estimate was used earlier in the paper [3].)

We remark that the analysis in this section was adapted to various singular situations in the paper [7], where the reader also can find references to other kinds of related asymptotic expansions.

The section is finished by verifying the 1 / 8-formula in Theorem 1.5.

### 5.1 Heat Kernel Estimate

Fix a number \(\vartheta <1\) (close to 1) and a smooth function \(\psi \) with \(\psi =1\) in \(D(0;\vartheta )\) and \(\psi =0\) outside *D*(0; 1). For given \(\zeta \in {{\mathbb {C}}}\) and \(\delta >0\), we define \(\chi \) by \(\chi (\omega )=\psi \left( (\omega -\zeta )/\delta \right) \). Then \(\chi =1\) in \(D\left( \zeta ;\vartheta \delta \right) \), \(\chi =0\) outside \(D\left( \zeta ;\delta \right) \), and the Dirichlet norm \(\left\| \bar{{\partial }}\chi \right\| \) depends only on \(\vartheta \). We sometimes write \(\chi _\zeta \) for \(\chi \).

*n*, and

*S*. We shall use the Hermitian analytic extension \(A(\zeta ,\eta )\) satisfying \(A(\zeta ,\zeta )=Q(\zeta )\) for \(\zeta \in \Omega \) (a neighborhood of

*S*). We assume that \(\delta _0\) is small enough that \(A(\zeta ,\eta )\) is defined whenever \(\left| {\zeta -\eta } \right| <2\delta _0\), \(\zeta \in S\).

*f*is a function supported in the domain of the function \({\mathbf {K}}_\eta ^\#\), we write

## Theorem 5.1

*C*such that \(\delta <\delta _n\) implies

The proof relies on the following lemma.

## Lemma 5.2

*u*is analytic in \(D(\zeta ;\delta )\), then

## Proof

*C*(independent of \(\zeta \),

*n*, and \(\delta \)) such that

## Proof of Theorem 5.1

### 5.2 Bergman Projection Estimate

Recall that \(L^2_\phi \) denotes the space of functions *f* normed by \(\left\| \,f\,\right\| _\phi ^{\,2}=\int \left| \,f\,\right| ^{\,2}e^{-\phi }\). We shall let \(A^2_\phi \) denote the subspace of \(L^2_\phi \) consisting of entire functions. We write \(\pi _\phi \) for the orthogonal (Bergman) projection \(L^2_\phi \rightarrow A^2_\phi \).

When \(\pi \) is the orthogonal projection of a Hilbert space onto a closed subspace, we denote by \(\pi ^\bot =I-\pi \) the complementary projection.

## Lemma 5.3

*C*such that

## Proof

*u*is a norm-minimal solution in \(L^2_{nQ}\) to the problem \({\bar{\partial }}u={\bar{\partial }}f\), where \(f=\chi _\zeta \cdot g_\zeta \). We shall prove that

*Q*near infinity. This gives \(\left\| \,v_0\,\right\| _{nQ}\le C\left\| \,v_0\,\right\| _{n\phi }\), and we have shown (5.4) with \(u=v_0\).

*u*is analytic in the disc \(D\left( \zeta ;\gamma /\sqrt{n}\right) \), so that Lemma 3.1 applies. We obtain that

*C*depends on \(\gamma \) and \(\Delta Q(\zeta )\). Combining with (5.5), we have shown (with a new

*C*)

The following result is just a restatement of part (ii) of Theorem 1.4.

## Theorem 5.4

*C*such that if \(\zeta \in S\) and \(\delta ={\text {dist}}(\zeta ,{\partial }S)\), then

## Proof

*C*). We can thus assume that \(\delta >\gamma /\sqrt{n}\). To this end, we put \(\ell =\vartheta ^{\,2}/2\). Then

### 5.3 An Exterior Estimate

*C*such that

*p*.

## Lemma 5.5

*p*is a regular boundary point at distance at least \(\delta \) from all singular boundary points, where \(\delta >0\) is independent of

*n*. There is then a constant \(C=C(\delta )\) such that whenever \(\zeta \in S^c\) and \(|z|\le \log n\), we have

## Proof

Let *V* be the harmonic continuation of \(\check{Q}|_{S^c}\) to a neighborhood of *p*. (We can choose the coordinate system so that \(\theta =0\) and the exterior normal derivative “\({\partial }/{\partial }n\)” (having nothing to do with the integer *n*) is simply \({\partial }/{\partial }x.\))

*x*. However, since

*p*is a regular point and \(Q=V\) on \({\partial }S\), we have \({{\partial }^2 M}/{{\partial }s^2}(p)=0\), where \({\partial }/{\partial }s\) denotes differentiation in the tangential direction. Adding this to the above Taylor expansion, using that \(({\partial }^2/{\partial }s^2+{\partial }^2/{\partial }n^2)M=4\Delta M=4\Delta Q\), we obtain, when \(|z|\le C\log n\),

It follows from the lemma that each limiting 1-point function \(R(z)=K(z,z)\) at a regular boundary point must satisfy \(R(z)\le Ce^{-2x^2}\), where \(x={\text {Re}}z\). This proves Theorem 1.4, part (i).

### 5.4 The 1 / 8-Formula

*S*is connected and that the boundary \({\partial }S\) is everywhere regular, so that the theory from [3] applies. Consider the class \({\mathcal C}_0\) of test functions \(f\in {C _0 ^\infty }({{\mathbb {C}}})\) with \(f=0\) on \({\partial }S\). For \(f\in {\mathcal C}_0\), we define functionals

*ds*is arclength measure on \({\partial }S\).

Now consider an arclength parametrization \(p=p(s)\) (\(0\le s\le s_0\)) of \({\partial }S\). Also denote by \(N_p\) the exterior unit normal at \(p\in {\partial }S\).

*s*,

*t*) where

*s*and

*t*are real parameters, related to the corresponding point \(\zeta =\zeta _n(s,t)\in {\mathcal N}_M\) by

*s*,

*t*)-system, the set \({\mathcal N}_M\) corresponds to a strip

*o*(1)-constant depends on

*M*.

*p*(

*s*) will be denoted by

*C*, \(c>0\) such that

*C*, \(C'\) such that

## Lemma 5.6

## Proof

*s*. The two cases are similar, so we just prove the inequality for \(h^*\).

Suppose that the set \(E_\alpha :=\left\{ h^*>\alpha \right\} \) has positive measure for some \(\alpha >0\). Take \(\varepsilon \in (0,1)\) to be fixed later.

Put \(\omega =(x-\delta ,x+\delta )\) and \(\tilde{\omega }=(x-\delta /2,x+\delta /2)\). Also pick a large number \(P>0\).

Now fix a test function \(f\in {\mathcal C}_0\) such that \(0\le {\partial }f/{\partial }n\le P\) on \({\partial }S\), \({\partial }f/{\partial }n=0\) outside \(\omega \), \({\partial }f/{\partial }n=P\) on \(\tilde{\omega }\), and \(\left\| \,f\,\right\| _1\le 1\).

*P*large enough, we can get a contradiction regardless of the values of \(\alpha >0\),

*m*, and \(C''\). The contradiction shows that \(\left| \,E_\alpha \,\right| =0\). \(\square \)

## 6 Translation Invariant Solutions

In this section, we study *t.i. * (translation invariant) limiting kernels and prove Theorems 1.6–1.9. The point of the translation invariance hypothesis is that we can interpret Ward’s equation as a convolution equation, which can be solved using Fourier analysis. If we rescale at a regular boundary point, the solution is fixed uniquely by this and our previously obtained a priori conditions.

### 6.1 The Convolution Representation of a Translation Invariant Limiting Kernel

Let \(\gamma (z)=\frac{1}{\sqrt{2\pi }}\, e^{-\,z^{\,2}/2}.\) In this section, we prove the following result:

## Lemma 6.1

Let \(K(z,w)=G(z,w)\Phi (z+\bar{w})\) be an arbitrary translation invariant limiting kernel. Then there exists a Borel function *f* with \(0\le f\le 1\) such that \(\Phi =\gamma *f\).

*translation invariant*(or

*t.i.*) if

## Lemma 6.2

\(\Psi \) is translation invariant if and only if \(\Psi (z,w)=\Phi (z+\bar{w})\) for some entire function \(\Phi \).

## Proof

*t.i.*, we define \(\Phi (z)=\Psi (z,0)\). We must prove that

*z*, both functions are analytic in \(\bar{w}\), and they coincide on the imaginary axis. \(\square \)

In the following, we fix any limiting holomorphic kernel \(L(z,w)=e^{z\bar{w}}\Psi (z,w).\) We shall apply Theorem 1.2 part (iii), which states that both *L* and \(\tilde{L}(z,w)=e^{z\bar{w}}(1-\Psi (z,w))\) are positive kernels.

*L*only on the imaginary axis, to conclude that for all finite subsets \(\{x_i\}_1^N\subset {{\mathbb {R}}}\) and all complex scalars \(\alpha _i\), we have

*V*(

*x*) is hence positive definite, and by Bochner’s theorem (e.g., [24]), it is the inverse Fourier transform of a positive measure \(\mu \). We have shown that

*f*and \(f_1\) are some non-negative Borel functions with \(f(x)+f_1(x)=1\). In particular, \(0\le f\le 1\). By this, the proof of Lemma 6.1 is complete. \(\square \)

### 6.2 Translation Invariant Solutions to Ward’s Equation

We shall now prove Theorem 1.6. Thus we shall find all solutions to Ward’s equation of the special form \(K(z,w)=G(z,w)\Phi (z+\bar{w})\), where \(\Phi =\gamma *f\) for some bounded Borel function *f*.

*V*(

*z*), which is translation invariant in the sense that \(V(z+it)=V(z)\) for all \(z\in {{\mathbb {C}}}\) and \(t\in {{\mathbb {R}}}\), satisfies \({\partial }V=\frac{1}{2} {\partial }_x V\). It is convenient to formulate the following reformulation of Ward’s equation in terms of \(J\):

## Lemma 6.3

*G*on \({{\mathbb {R}}}\), such that

## Proof

Set \(G(x)=P(x/2)\) and \(L(x)=D(x/2)\) in Lemma 4.9, where we recall that *D*(*z*) is defined by the integral (4.6). \(\square \)

We will need two elementary lemmas.

## Lemma 6.4

## Proof

*n*is odd, the zeroth Fourier coefficient of \((te^{i\theta }+se^{-i\theta })^n\) vanishes, while if

*n*is even, then

## Lemma 6.5

## Proof

*f*, the restriction \(J\) of \(\Phi \) to \({{\mathbb {R}}}\) has the structure of the usual convolution

*f*is regarded as a tempered distribution. This is well defined, since \(\hat{\gamma }=\sqrt{2\pi }\,\gamma \) is a Schwartz test function.

*L*(

*x*) in (6.2). Using (6.5) and Lemma 6.5, we have

*L*(

*x*), we get

*L*(

*x*), we have arrived at the following result:

## Lemma 6.6

*G*such that

We now prove Theorem 1.6.

*g*be a continuous function on \({{\mathbb {R}}}\) such that \(g'=f-1\); this determines

*g*up to a constant. Let us define \(G=g*\gamma \). Then

*g*. We can rewrite (6.6) in the form

*E*is a closed set, and the complement \(E^c={{\mathbb {R}}}\setminus E\) can be written as a countable union of disjoint open intervals \(I_j\). On each \(I_j\), we have \(f=0\) and \(g'=-1\) almost everywhere. Since \(g=0\) at the endpoints, none of the intervals can be finite. Hence

*E*is connected. Differentiating the relation \(fg=0\) and using \(g'=f-1\), we obtain that \(f=f^{\,2}\) when \(f\ne 0\). Hence \(f={\mathbf {1}}_E\) almost everywhere. We have shown that \(\Phi \) is representable in the form

### 6.3 Translation Invariant Limiting Kernels at Regular Boundary Points

In this subsection, we prove Theorem 1.7.

## Theorem 6.7

If the limiting kernel *K* at a regular boundary point is translation invariant, then \(K(z,w)=G(z,w)\Phi (z+\bar{w})\), and there is some *a* such that \(\Phi = \gamma * {\mathbf {1}}_{(-\infty ,a)}.\) Furthermore, if \(R(z)=\Phi (z+\bar{z})\) satisfies \(\int _{{\mathbb {R}}}t\cdot (R(t)-{\mathbf {1}}(t))\, dt=\frac{1}{8},\) then \(\Phi =F\) is the plasma function.

## Proof

By Lemmata 6.1 and 6.2, \(\Psi (z,w) = \Phi (z + {\bar{w}})\), where \(\Phi = \gamma * f\) for some bounded function *f* with \(0 \le f \le 1.\) By Theorem 1.4, we know that \(R(x)\rightarrow 1\) as \(x\rightarrow -\infty \) and \(R(x)\rightarrow 0\) as \(x\rightarrow +\infty \). Moreover, by Theorem 1.6, we can write \(\Phi =\gamma *{\mathbf {1}}_{(-\infty ,a)}\) for some \(a\in {{\mathbb {R}}}\). For the last statement, we must prove that \(a=0\).

*X*is a standard normal random variable. Hence the integral in the right-hand side can be written

*t.i.*limiting kernel

*R*. We know that \(\int _{{\mathbb {R}}}t(R(t)-{\mathbf {1}}(t))\, dt=\frac{1}{8}\). As we observed above, we can also write \(R(x)=\gamma *{\mathbf {1}}_{(-\infty ,a)}(2x)=F(2x-a)\) for some \(a\in {{\mathbb {R}}}\), and we must prove that \(a=0\). However, by (6.7),

### 6.4 Radially Symmetric Potentials

We now prove Theorem 1.8. We start with a simple lemma.

## Lemma 6.8

Assume that *Q* is radially symmetric. Fix a point \(p\in {\partial }S\), and rescale in the outwards normal direction (see (1.4)). Then every limiting kernel in Theorem 1.1 takes the form \(K=G\Psi \), where \(\Psi (z,w)=\Phi (z+\bar{w})\) is translation invariant.

## Proof

*Q*is radially symmetric and that the droplet

*S*is connected; thus it is either a disc or an annulus. If \(p\in {\partial }S\) is a boundary point, then the outer normal \(N_p\) is simply a multiple of

*p*, \(N_p=\pm p/|p|\). We can assume that \(|p|=1\) and \(N_p=p\). Let us write

*p*. The radial symmetry of

*Q*implies that \(R_{n,p}=R_{n,pe^{i\theta }}\) for all real \(\theta \). From this we conclude that the almost-everywhere convergence in Theorem 1.5 must hold pointwise; i.e., if \(R=\lim R_{n_k,p}\) is any limiting 1-point function, then

*R*corresponds to a

*t.i.*limiting kernel \(K(z,w)=G(z,w)\Phi (z+\bar{w})\). An application of Theorem 6.7 now shows that \(\Phi =F\) is the plasma function. The proof of Theorem 1.8 is finished.

### 6.5 Translation Invariant Solutions to the Mass-One Equation.

We now prove Theorem 1.9.

Let \(\Phi \) be an entire function of the form \(\Phi =\gamma *f\), where *f* is some bounded function.

*f*” lead to the equation

*E*is some measurable set of positive measure, and \(\Phi ={\mathbf {1}}_E*\gamma \). The proof of Theorem 1.9 is finished.

## 7 Ward’s Equations in Some Other Settings

In this section, we will rescale Ward identities and derive the corresponding equations in several different settings.

### 7.1 Ward’s Equation at the Hard Edge of the Spectrum

For simplicity, we shall restrict our discussion to the hard edge Ginibre ensemble; we refer to [5] for a discussion of more general hard edge ensembles.

Let \(\{\zeta _j\}_1^n\) be the hard-edge Ginibre process, and rescale about the boundary point \(p=1\) to obtain the boundary process \(\Theta _n=\{z_j\}_1^n\), where \(z_j=\sqrt{n}\left( \zeta _j-1\right) .\)

## Theorem 7.1

## Proof

*o*(1) converges to zero uniformly on compact subsets of \({{\mathbb {L}}}\).

In order to prove this, it is convenient to consider the Ginibre potential \(Q(\zeta )=\left| {\,\zeta +1\,} \right| ^{\,2}\) which has the droplet \(S=\{\,\left| {\,\zeta +1\,} \right| \le 1\,\}\). We rescale about the boundary point \(p=0\) via \(z=\sqrt{n}\,\zeta .\)

Fix a number \(\varepsilon >0\). Write \(U:=S\cap D(0;\varepsilon )\), and consider test functions \(\psi \) supported in the dilated set \(\sqrt{n}\cdot U\). As in the free case, we define \(\psi _n\left( \zeta \right) :=\psi \left( z\right) \). Since \(Q^S=Q\) in the set *U* where \(\psi _n\) is supported, the same arguments used in the free boundary case remain valid (cf. Subsection 4.2). The only difference is that the dilated domains \(\sqrt{n}\cdot U\) will, in our present case, increase to the open left half-plane \({{\mathbb {L}}}\). Hence we deduce Ward’s equation (7.1) for \(z\in {{\mathbb {L}}}\) precisely as before.

By Theorem 2.3, we have convergence \(R_n\rightarrow R\) and \(C_n\rightarrow C\) locally uniformly in \({{\mathbb {L}}}\) and boundedly almost everywhere in \({{\mathbb {C}}}\). It follows that we can pass to the limit in (7.1). \(\square \)

## Corollary 7.2

### 7.2 Ward’s Equation at Bulk Singularities and Mittag–Leffler Fields

Let us weaken our standing assumptions on the potential *Q*. We still require real-analyticity in a neighborhood of *S*, but now allow that \(\Delta Q=0\) at isolated points in the bulk of *S*. A point \(p\in {\text {Int}}S\) such that \(\Delta Q(p)=0\) will be called a *bulk singularity*.

Assume that \(p=0\) is a bulk singularity, and let \(\{\zeta _j\}_1^n\) be the point process corresponding to *Q*. The effect of the bulk singularity is to repel the particles away from it.

*p*. For instance, if \(\Delta Q(\zeta )=ax^2+by^2+O(|\zeta |^3)\) as \(\zeta =x+iy\rightarrow 0\), where

*a*and

*b*are positive constants, then the local behavior of the system \(\{\zeta _j\}\) near 0 will depend on

*a*as well as

*b*. Let us consider the symmetric case when \(a=b=1\), or more generally, that there is a number \(\lambda \ge 1\) such that

*Q*to be real-analytic, we should of course assume that \(\lambda \) be an integer. However, the condition of real-analyticity is important only in a neighborhood of the boundary, e.g., in connection with Sakai’s theory. In the bulk, it suffices to assume \(C^2\)-smoothness. Thus we can in fact choose \(\lambda \) as an arbitrary real constant \(\ge 1\). Note that \(\lambda =1\) is the well-known case of an ordinary “regular” bulk point, in which case we know that the usual Ginibre point field arises. We may thus assume that \(\lambda >1\).

## Example

In the next theorem, we consider Ward’s equation at \(p=0\) for the potential \(Q_\lambda (\zeta )=\left| {\,\zeta \,} \right| ^{\,2\lambda }\). To this end, we introduce the Berezin kernel rescaled about 0 on the scale (7.3), i.e., \(B_{n}(z,w):={\left| {\,K_n(z,w)\,} \right| ^{\,2}}/ {K_n(z,z)}.\)

## Theorem 7.3

*B*is a solution to the Ward’s equation

## Proof

It is easy to see that \(M_\lambda \) is of exponential type \(\lambda \). This implies that the kernel *K* is uniformly bounded. Existence and uniqueness of a point field \(ML_\lambda \) with the given properties now follows, via Lenard’s theory, from the convergence of intensities in the preceding example.

To this end, fix a test function \(\psi \), and let \(\psi _n\left( \zeta \right) =\psi \left( z\right) \), where \(z=n^{\,1/(2\lambda )}\,\zeta \).

*k*-point function of the system \(\{\zeta _j\}_1^n\). The rescaling \(z_j=n^{1/(2\lambda )}\zeta _j\) then implies that the

*k*-point function of the rescaled system \(\{z_j\}_1^n\) is

*k*is a positive integer. Rescaling by a suitable factor proportional to \(n^{-1/2k}\), one deduces the asymptotic relation

An equally interesting generalization is obtained by allowing the potential to have a weak logarithmic singularity at the origin. This corresponds to the microscopic study of a particle system on a Riemann surface, close to a conical singularity, or alternatively, to the study of the microscopic effect of insertion of a point charge. This possibility is considered in the papers [4, 8] and will also be the subject of a forthcoming investigation.

### 7.3 Ward’s Equation and the Mass-One Equation for \(\beta \)-Ensembles

*Q*satisfying the standing assumptions in Subsection 1.1, and fix a number \(\beta >0\). Let us consider the probability measure on \({{\mathbb {C}}}^n\) defined by

Let \({\mathbf R}_{n,k}^{\,\beta }\) denote the corresponding *k*-point function. The following version of Ward’s identity is proved exactly as in the case \(\beta =1\).

## Theorem 7.4

*Q*is \(C^2\)-smooth near \({\text {supp}}\psi \), then \({{\mathbb {E}}}_n^{\,\beta }W_n^+[\psi ]=0\).

*p*: \(z_j=e^{\,-i\theta }\sqrt{n\Delta Q(p)}\,\left( \zeta _j-p\right) \). We denote by \(\Theta _n^{\,\beta }:=\{z_j\}_1^n\) the rescaled process and write \(R_{n,k}^{\,\beta }(z_1,\ldots ,z_k):={\mathbf R}_{n,k}^{\,\beta }(\zeta _1,\ldots ,\zeta _k)\) for the joint intensities. We also define the

*Berezin kernel*of the process \(\Theta _n^{\,\beta }\) by

Rescaling the Ward identity as in Subsection 4.2, we obtain the following result.

## Theorem 7.5

*p*belongs to some neighborhood of

*S*in which

*Q*is strictly subharmonic and \(C^2\)-smooth, then

## Proposition 7.6

Suppose that \(B^{\,\beta }\) solves (7.7). Then the kernel *B* in (7.8) solves Ward’s equation (with \(\beta =1\)).

We do not know whether the (presumptive) kernels \(B^{\,\beta }\) would be non-negative, so speaking about “mass-one” could possibly be misleading. However, if we assume that \(\int _{{\mathbb {C}}}B^{\,\beta }(z,w)\, dA(w)=1\), then the corresponding kernel *B* in (7.8) satisfies the “mass-\(\beta \) equation”: \(\int _{{\mathbb {C}}}B(u,v)\, dA(v)=\beta \).

As we mentioned before, in the case \(\beta =1\), Ward’s equation is “closed” by analytic continuation. We don’t know if we can consider equation (7.8) closed if \(\beta \ne 1.\)

The study of boundary profiles \(R^\beta (x)\) for a given \(\beta >1\) is of physical relevance, in connection to the Hall effect; see the paper [13].

## 8 Concluding Remarks

In Subsection 8.1, we explain how the boundary kernel \(K = GF\) in the Ginibre case can be related to asymptotics of section of the exponential function. In Subsection 8.2, we will mention some connections to the theory of Hilbert spaces of entire functions and to the theories of certain special functions. In Subsection 8.3, we comment on the nature of the mass-one equation and Ward’s equation in the general (non-translation-invariant) case, relating those equations to harmonic analysis on the Heisenberg group.

### 8.1 Sections of Power Series

*section*of an entire function \(f(\zeta )=\sum _{j=0}^\infty a_j\zeta ^{\,j}\), we here simply mean a partial sum

*w*

*except*for

*w*in a fixed neighborhood of 1. This gap was later closed, and the following result ensued. Consider the rescaled section

*F*is the plasma function (0.3). We are unsure concerning whom should be credited for the convergence in (8.1) when \(f(\zeta )=e^\zeta \). However, the book [15, Theorem 1] contains a statement valid for more general

*f*, and the appendix in [10] contains a detailed convergence result for the case at hand.

The convergence in (8.1) has been proved for the sections corresponding to more general entire functions. In the monograph [15], the authors consider the Mittag–Leffler function \(E_{1/\lambda }\) as well as a class denoted “\(\mathcal {L}\)-functions.” More recently, this kind of convergence has been used in the papers [29, 39] (it is called “Newman–Rivlin asymptotics” in [39]).

To interpret the above results in terms of our Theorem 1.8, one chooses a suitable radially symmetric potential *Q*. For example, one chooses \(Q(\zeta )=E_{1/\lambda }(|\,\zeta \,|^{\,2})\) in case of the Mittag–Leffler function alluded to above. Expressing the kernel \({\mathbf {K}}_n\) in terms of the orthogonal polynomials (as in (1.3)) and rescaling about a boundary point of the droplet, one can apply Theorem 1.8 and recover the asymptotic behavior of the sections.

### 8.2 The Mass-One Equation and Hilbert Spaces of Entire Functions

*Hermitian*matrices are related to certain specific

*de Branges*spaces \({\mathcal B}(E)\) of entire functions. See [14] for the definition of these spaces. In particular, the sine-kernel describing the spacing of eigenvalues in the bulk is the restriction to \({{\mathbb {R}}}^2\subset {{\mathbb {C}}}^2\) of the reproducing kernel of the Paley–Wiener space, i.e., the space \({\mathcal B}(E)\) where \(E(z)=e^{-i\pi z}\). Moreover, the Airy kernel

The appearance of de Branges spaces in the context of Hermitian random matrices is quite natural given the fact that orthogonal polynomials on the real line can be related to a second order one-dimensional self-adjoint spectral problem.

The Hilbert spaces \({\mathcal H}\) of entire functions arising in the random normal matrix theory are not of de Branges type, and we are not sure about their spectral interpretation. Nevertheless, we will use the term “spectral measure”: \(\mu \) is a spectral measure for \({\mathcal H}\) if \({\mathcal H}\) sits isometrically in \(L^2(\mu )\).

## Lemma 8.1

- (i)The kernel \(K=G\Psi \) satisfies the mass-one equation; i.e.,$$\begin{aligned} \int e^{\,-\,|\,w\,|^{\,2}}\left| {\,\Psi (z,z+w)\,} \right| ^{\,2}\, dA(w)=\Psi (z,z),\quad z\in {{\mathbb {C}}}.\end{aligned}$$(8.2)
- (ii)
The holomorphic kernel \(L(z,w)=e^{\,z\bar{w}}\Psi (z,w)\) is the reproducing kernel of some Hilbert space \({\mathcal H}\) with spectral measure \(d\mu (z):=e^{\,-\,|\,z\,|^{\,2}}\, dA(z)\).

*K*.

## Proof

*L*is the reproducing kernel for a Hilbert space with spectral measure \(d\mu (z)=e^{\,-\,|\,z\,|^{\,2}}\, dA(z)\) if and only if

*L*gives rise to a reproducing kernel as in (ii), then

*K*is the reproducing kernel of the subspace \({\mathcal W}=\left\{ \,f;\, f(z)=g(z)e^{\,-\,|\,z\,|^{\,2}/2},\, g \in {\mathcal H}\,\right\} \) of \(L^2\).

Consider the linear operator *T* on \(L^2\) with kernel *K*. Then *T* is an orthogonal projection, and it is locally trace class. (That an operator *T* on \(L^2\) is “locally trace class” means that the operator \(T_B\) on \(L^2\) defined by \(T_B(f)=T({\mathbf {1}}_B f)\) is trace class for every compact set \(B\subset {{\mathbb {C}}}\).) By a theorem of Soshnikov ( [34, Theorem 3]), the conditions above guarantee that *K* is the correlation kernel of a unique random point field in \({{\mathbb {C}}}\). \(\square \)

*entire*functions of class \(L^2(\mu )\). It follows from general facts for reproducing kernels that the Hilbert space \({\mathcal H}\) in (ii) is the closed linear span

*F*is the plasma function (0.3).

*H*is the hard edge function (0.5). The fact that the last span consists of entire functions requires a compactness property in the hard edge situation, which will be established in the paper [5].

It would be interesting to describe the above spaces in more constructive terms (e.g., similar to de Branges theory). It would also be interesting to know the meaning of Ward’s equation for the spaces \({\mathcal H}\). (By Lemma 8.1, the mass-one equation is a statement about spectral measures.)

In general, we expect that a kernel *L* arising from rescaling in a (free-boundary) ensemble should be the Bergman kernel of a subspace of a “generalized Fock–Sobolev space”; see [7, 8] for results in this direction.

### 8.3 Twisted Convolutions

We finally show that, without the hypothesis of translation invariance, Ward’s equation takes the form of a so-called twisted convolution equation, known from Weyl’s calculus for pseudodifferential operators. A solution of this equation, coupled with the a priori estimates above, could plausibly lead to a proof of the hypothesis of translation invariance (say, at a regular boundary point). This thread will be taken up elsewhere.

*f*,

*g*defined on \({{\mathbb {C}}}\), we define the

*twisted convolution*\(f\star g\) by

*f*such that

By polarizing in the Fourier inversion formula, we obtain an analogue of the identity (6.4):

## Lemma 8.2

Now define \(R_0=1-R\), and assume that we can represent \(R_0\) in a similar way to (8.4) \(\hat{R}_0=\hat{g}\cdot \Gamma ,\) where *g* is a suitable function. Lemma 8.2 then allows us to rewrite the mass-one equation and Ward’s equation as follows.

*Mass-one equation.*

*Ward’s equation.*There exists a smooth function \(P_0\) such that \({\bar{\partial }}P_0=R_0\) and

## Notes

### Acknowledgements

We thank Alexei Borodin and Misha Sodin for their interest. We also thank Aron Wennman, Seong-Mi Seo, Hee-Joon Tak, and Sungsoo Byun for careful reading and much appreciated help with improving this manuscript.

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