Abstract
Consider the subspace \({{{\mathscr {W}}}_{n}}\) of \(L^2({{\mathbb {C}}},dA)\) consisting of all weighted polynomials \(W(z)=P(z)\cdot e^{\frac{1}{2}nQ(z)},\) where P(z) is a holomorphic polynomial of degree at most \(n1\), \(Q(z)=Q(z,{\bar{z}})\) is a fixed, realvalued function called the “external potential”, and \(dA=\tfrac{1}{2\pi i}\, d{\bar{z}}\wedge dz\) is normalized Lebesgue measure in the complex plane \({{\mathbb {C}}}\). We study large n asymptotics for the reproducing kernel \(K_n(z,w)\) of \({{\mathscr {W}}}_n\); this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of \(\hat{{{\mathbb {C}}}}\setminus S\) containing \(\infty \), leaving aside such cases which are at this point wellunderstood. For the Ginibre kernel, corresponding to \(Q=z^2\), we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate fullplane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tailkernel approximation” and summing by parts. In the offdiagonal case \(z\ne w\) when both z and w are on the boundary \({\partial }U\), we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic
where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space \(H^2_0(U)\) of analytic functions on U vanishing at infinity, equipped with the norm of \(L^2({\partial }U,dz)\). Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.
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1 Introduction
1.1 The Ginibre ensemble
Recall that the standard (complex) Ginibre ensemble [40, 45, 56, 64, 68] is the determinantal pointprocess \(\{z_j\}_1^n\) in the complex plane \({{\mathbb {C}}}\) with kernel
To arrive at this kernel, we are prompted to equip \({{\mathbb {C}}}\) with the background measure
The law of \(\{z_j\}_1^n\) is the Gibbs measure
where \(dA_n=(dA)^{\otimes n}\) is the normalized Lebesgue measure on \({{\mathbb {C}}}^n\). (The combinatorial factor 1/n! accounts for the fact that elements \((z_j)_1^n\in {{\mathbb {C}}}^n\) are ordered sequences, while configurations \(\{z_j\}_1^n\) are unordered.)
The expected number of particles which fall in a given Borel set E is
and if \(f(z_1,\ldots ,z_k)\) is a compactly supported Borel function on \({{\mathbb {C}}}^k\) where \(k\le n\), then
where the kpoint function \(R_{n,k}(w_1,\ldots ,w_k)=\det (K_n(w_i,w_j))_{i,j=1}^k\). We reserve the notation
for the 1point function.
The circular law (e.g. [17, 45]) states that \(\tfrac{1}{n}R_n(z)\) converges as \(n\rightarrow \infty \) to the characteristic function \({\textbf{1}}_S(z)\), where S (the droplet) is the closed unit disc \(\{z\le 1\}\). More refined asymptotic estimates may be found in [6, 12, 26, 27, 38, 41, 54, 68], for example.
We shall here study the case when \(z{\bar{w}}1\ge \eta \) for some \(\eta >0\) and deduce asymptotics for \(K_n(z,w)\) using techniques which hark back to Szegő’s work [72] on the distribution of zeros of partial sums of the Taylor series of the exponential function. With a suitable interpretation, the asymptotic turns out generalize to to a large class of random normal matrix ensembles. In addition we shall find that the socalled Szegő kernel emerges in the offdiagonal boundary asymptotics. For those reasons we shall refer to a group of asymptotic results below as “Szegő type”.
The complete asymptotic picture of (1.1) is intimately connected with the Szegő curve
We define the exterior Szegő domain \(E_{\textrm{sz}}\) to be the unbounded component of \({{\mathbb {C}}}\setminus \gamma _{\textrm{sz}}\), i.e.,
(See Fig. 1)
1.1.1 Szegő type asymptotics for the Ginibre kernel
Three principal cases emerge, depending on the location of the product \(z{\bar{w}}\).

(i)
If \(z{\bar{w}}\in {{\mathbb {C}}}\setminus (E_{\textrm{sz}}\cup \{1\})\) we have bulk type asymptotic in the sense that
$$\begin{aligned} K_n(z,w)=ne^{nz{\bar{w}}\frac{n}{2} z^2\frac{n}{2} w^2}\cdot (1+O(n^{\frac{1}{2}})). \end{aligned}$$(Cf. Sect. 2.2 for more about this.)

(ii)
If \(z{\bar{w}}\) is in a microscopic neighbourhood of \(z{\bar{w}}=1\), then (1.1) has a wellunderstood errorfunction asymptotic given in [12, Subsection 2.2]. Further results in this direction can be found in [26, 54, 73], for example.

(iii)
If \(z{\bar{w}}\in E_{\textrm{sz}}\), it turns out that (1.1) has a third kind of asymptotic, which we term exterior type. This is our main concern in what follows, and we immediately turn our focus on it.
Theorem 1.1
Suppose that \(z{\bar{w}}\in E_{\textrm{sz}}\) and let \(K_n(z,w)\) be the Ginibre kernel (1.1). Then as \(n\rightarrow \infty \)
The Oconstant is uniform provided that \(\zeta =z{\bar{w}}\) remains in a compact subset of \(E_{\textrm{sz}}\); the correction term \(\rho _j(\zeta )\) is a rational function having a pole of order 2j at \(\zeta =1\) and no other poles in the extended complex plane \(\hat{{{\mathbb {C}}}}={{\mathbb {C}}}\cup \{\infty \}\); the first one is given by
and the higher \(\rho _j(\zeta )\) can be computed by a recursive procedure based on (1.34), (1.35) below.
In the case when \(\zeta =z{\bar{w}}\) belongs to the sector \(\arg (\zeta 1)<\tfrac{3\pi }{4}\), the result can alternatively be deduced by writing the kernel as a product involving an incomplete gammafunction and appealing to an asymptotic result due to Tricomi [75]. Our present approach (found independently) is quite different and has the advantage of leading to the precise domain \(E_{\textrm{sz}}\) where the same asymptotic formula applies. See Sect. 1.5 for further details.
For \(k=0\), Theorem 1.1 implies that
Now assume that both z and w are on the unit circle \({{\mathbb {T}}}={\partial }S=\{z=1\}.\) In this case the function \(c_n(z,w)=z^n{\bar{w}}^n=\frac{z^n}{w^n}\) is a cocycle, which may be canceled from the kernel (1.5) without changing the value of the determinant (1.2). Therefore, using the symbol “\(\sim \)” to mean “up to cocycles”, (1.5) implies
where S(z, w) is the (exterior) Szegő kernel
Let \({{\mathbb {D}}}_e=\{z>1\}\cup \{\infty \}\) be the exterior disc and \(d\theta =dz\) the arclength measure on \({{\mathbb {T}}}\). Consider the Hardy space \(H^2_0({{\mathbb {D}}}_e)\) of analytic functions \(f:{{\mathbb {D}}}_e\rightarrow {{\mathbb {C}}}\) which vanish at infinity, equipped with the norm of \(L^2({{\mathbb {T}}},d\theta )\). The kernel S(z, w) is the reproducing kernel of \(H^2_0({{\mathbb {D}}}_e)\).
Let us now consider the Berezin kernel rooted at a point \(z\in {{\mathbb {C}}}\),
It is a household fact that if \(z\in {{\mathbb {T}}}\), then \(K_n(z,z)=\frac{1}{2} \, n\cdot (1+o(1)).\)
(Proof: \(K_n(z,z)=n\cdot {\mathbb {P}}(\{X_n\le n\})\) where \(X_n\) is a Poisson random variable with intensity n. Since \((X_nn)/\sqrt{n}\) converges in distribution to a standard normal, \({\mathbb {P}}(\{X_n\le n\})\rightarrow \frac{1}{2}\) as \(n\rightarrow \infty \).)
It follows that if \(z,w\in {{\mathbb {T}}}\) and \(z\ne w\), then
It is interesting to compare (1.9) with the case when \(z\in {\text {Int}}S\); then \(B_n(z,w)\) decays exponentially in n by the heatkernel estimate in Sect. 2.2. (Alternatively, by results in [9].)
The moral is that, in the offdiagonal case \(z\ne w\), the magnitude of \(K_n(z,w)\) is exceptionally large when both z and w are on the boundary \({{\mathbb {T}}}\), compared with any other kind of configuration. (Some heuristic explanations for this kind of behaviour are sketched below in Sect. 1.3.)
1.1.2 Gaussian convergence of Berezin measures
It is natural to regard the Berezin kernel (1.8) as the probability density of the Berezin measure \(\mu _{n,z}\) rooted at z,
It is shown in [9, Section 9] that if \(z\in {{\mathbb {D}}}_e\) then the measures \(\mu _{n,z}\) converge weakly to the harmonic measure relative to \({{\mathbb {D}}}_e\) evaluated at z, \(d\omega _z(\theta )=P_z(\theta )\, d\theta \) where \(P_z(\theta )\) is the (exterior) Poisson kernel
We will denote by \(d\gamma _n\) the following Gaussian probability measure on \({{\mathbb {R}}}\)
(Here and throughout, “\(d\ell \)” is Lebesgue measure on \({{\mathbb {R}}}\).)
It is also convenient to represent points w close to \({{\mathbb {T}}}\) in “polar coordinates”
As a consequence of the kernel asymptotic in Theorem 1.1, we obtain the following result.
Corollary 1.2
Fix a point \(z\in {{\mathbb {D}}}_e\) and an arbitrary sequence \((c_n)_1^\infty \) of positive numbers with
Then for w in the form (1.12), we have the Gaussian approximation
where \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \) uniformly for w in the belt \(N({{\mathbb {T}}},c_n):=\{\ell \le c_n\}.\)
Here and henceforth, a sequence of functions \(f_n:E_n\rightarrow {{\mathbb {C}}}\) is said to converge uniformly to 0 if there is a sequence \(\epsilon _n\rightarrow 0\) such that \(f_n<\epsilon _n\) on \(E_n\) for each n.
Remark
The approximating measures \(d{\tilde{\mu }}_{n,z}(\theta ,\ell )=P_z(\theta )\cdot \gamma _n(\ell )\, d\theta d\ell \) are probability measures on \({{\mathbb {T}}}\times {{\mathbb {R}}}\) which assign a mass of \(O(e^{nc_n^{\,2}})\) to the complement of \(N({{\mathbb {T}}},c_n)\). The condition that \(nc_n^2\rightarrow \infty \) insures that \(\mu _{n,z}{\tilde{\mu }}_{n,z}\rightarrow 0\) in the sense of measures on \({{\mathbb {C}}}\). This justifies the Gaussian approximation picture, as exemplified in Fig. 2.
1.2 Notation and potential theoretic setup
In order to generalize beyond the Ginibre ensemble, we require some notions from potential theory; cf. [69].
We are about to write down rather a dry list of definitions and generally useful facts; the reader may skim it to his advantage.
We begin by fixing a lower semicontinuous function \(Q:{{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\cup \{+\infty \}\) which we call the external potential. (The Ginibre ensemble corresponds to the special choice \(Q(z)=z^2\).)
We assume that Q is finite on some set of positive capacity and that
Further conditions are given below.
Given a compactly supported Borel probability measure \(\mu \), we define its Qenergy by
where \(\mu (Q)\) is short for \(\int Q\, d\mu \).
It is wellknown [69] that there exists a unique equilibrium measure \(\sigma =\sigma _Q\) of unit mass which minimizes \(I_Q[\mu ]\) over all compactly supported Borel probability measures on \({{\mathbb {C}}}\). The support of \(\sigma \) is denoted by \(S=S[Q]={\text {supp}}\sigma ,\) and is called the droplet. Perhaps even more central to this work is the exterior component containing \(\infty \),
The boundary of U is called the outer boundary of S and is written \(\Gamma ={\partial }U.\)
We now introduce four standing assumptions (1)(4).

(1)
S is connected and Q is \(C^2\) smooth in a neighbourhood of S and realanalytic in a neighbourhood of \(\Gamma \).
This assumption has the consequence that the equilibrium measure \(\sigma \) is absolutely continuous and has the structure \(d\sigma ={\textbf{1}}_S\cdot \Delta Q\, dA,\) where \(\Delta ={\partial }\bar{\partial }=\tfrac{1}{4} (\tfrac{{\partial }^2}{{\partial }x^2}+\tfrac{{\partial }^2}{{\partial }y^2})\) is the normalized Laplacian.
We are guaranteed that \(\Delta Q\ge 0\) on S; we will require a bit more:

(2)
\(\Delta Q(z)>0\) for all \(z\in \Gamma \).
Let \({\textrm{SH}}_1(Q)\) denote the class of all subharmonic functions s(z) on \({{\mathbb {C}}}\) which satisfy \(s\le Q\) on \({{\mathbb {C}}}\) and \(s(z)\le \log z^2+O(1)\) as \(z\rightarrow \infty \). We define the obstacle function \({\check{Q}}(z)\) to be the envelope
Clearly \({\check{Q}}(z)\) is subharmonic and grows as \(\log z^2+O(1)\) as \(z\rightarrow \infty \). Furthermore, \({\check{Q}}(z)\) is \(C^{1,1}\)smooth on \({{\mathbb {C}}}\), i.e., its gradient is Lipschitz continuous.
Denote by \(S^*=\{z\, ;\, Q(z)={\check{Q}}(z)\}\) the coincidence set for the obstacle problem. In general we have the inclusion \(S\subset S^*\) and if p is a point of \(S^*\setminus S\) then there is a neighbourhood N of p such that \(\sigma (N)=0\). We impose:

(3)
\(U\cap S^*\) is empty.
Write \(\chi :{{\mathbb {D}}}_e\rightarrow U\) for the unique conformal mapping normalized by the conditions \(\chi (\infty )=\infty \) and \(\chi '(\infty )>0\). A fundamental theorem due to Sakai [70] implies that \(\chi \) extends analytically across \(\Gamma \) to some neighbourhood of the closure \({\text {cl}}{{\mathbb {D}}}_e\). (Details about this application of Sakai’s theory are found in Sect. 3.1 below.) Thus \(\Gamma \) is a Jordan curve consisting of analytic arcs and possibly finitely many singular points where the arcs meet. We shall assume:

(4)
\(\Gamma \) is nonsingular, i.e., \(\chi \) extends across \({{\mathbb {T}}}\) to a conformal mapping from a neighbourhood of \({\text {cl}}{{\mathbb {D}}}_e\) to a neighbourhood of \({\text {cl}}U\).
In the following we denote by \(\phi =\chi ^{1}\) the inverse map, taking a neighbourhood of \({\text {cl}}U\) conformally onto a neighbourhood of \({\text {cl}}{{\mathbb {D}}}_e\), and obeying \(\phi (\infty )=\infty \) and \(\phi '(\infty )>0\). We denote by \(\sqrt{\phi '}\) the branch of the squareroot which is positive at infinity.
1.2.1 Class of admissible potentials
Except when otherwise is explicitly stated, all external potentials Q used below are lower semicontinuous functions \({{\mathbb {C}}}\rightarrow {{\mathbb {R}}}\cup \{+\infty \}\), finite on some set of positive capacity, satisfying the growth condition (1.14) and the four conditions (1)(4).
1.2.2 Auxiliary functions
For a given admissible potential Q, we consider the holomorphic functions \({{\mathscr {Q}}}(z)\) and \({{\mathscr {H}}}(z)\) on a neighbourhood of \({\text {cl}}U\) which obey
and which satisfy \({\text {Im}}{{\mathscr {Q}}}(\infty )={\text {Im}}{{\mathscr {H}}}(\infty )=0\).
We shall also frequently use the function V given by
It is useful to note the identity
where K is a fixed compact subset K of the bounded component \({\text {Int}}\Gamma \) of \({{\mathbb {C}}}\setminus \Gamma \).
To realize (1.19) it suffices to note that the harmonic functions on the left and right hand sides agree on \(\Gamma \) and grow like \(\log z^2+O(1)\) near infinity, so (1.19) follows by the strong version of the maximum principle (e.g. [44]).
1.2.3 The Szegő kernel
Let \(H_0^2(U)\) be the Hardy space of holomorphic functions \(f:U\rightarrow {{\mathbb {C}}}\) which vanish at infinity and are squareintegrable with respect to arclength: \(\int _\Gamma f(z)^2\,dz<\infty \). We equip \(H^2_0(U)\) with the inner product of \(L^2(\Gamma ,dz)\) and observe that the functions \(\psi _j(z)=\frac{1}{\sqrt{2\pi }}\frac{\sqrt{\phi '(z)}}{\phi (z)^j}\) (\(j\ge 1\)) form an orthonormal basis for \(H^2_0(U)\). The reproducing kernel for \(H^2_0(U)\) is thus
We shall refer to S(z, w) as the Szegő kernel associated with \(\Gamma \) (or U).
Many interesting properties of the Szegő kernel can be found in Garabedian’s thesis work [43] and in the book [23]. A different natural way to define \(H^p\)spaces over general domains is discussed in e.g. [36, Section 10].
1.2.4 The reproducing kernel
Let Q be an admissible potential and consider the space \({{\mathscr {W}}}_n={{\mathscr {W}}}_n(Q)\) consisting of all weighted polynomials W of the form
where P is a holomorphic polynomial of degree at most \(n1\). We equip \({{\mathscr {W}}}_n\) with the usual norm in \(L^2({{\mathbb {C}}},dA)\) and denote by \(K_n(z,w)\) the corresponding reproducing kernel.
We follow standard conventions concerning reproducing kernels [15]; we write \(K_{n,z}(w)=K_n(w,z)\) and note that the element \(K_{n,z}\in {{\mathscr {W}}}_n\) is characterized by the reproducing property:
for all \(W\in {{\mathscr {W}}}_n\) and all \(z\in {{\mathbb {C}}}\).
We shall frequently use the formula
where \(\{W_{j,n}\}_{j=0}^{n1}\) is any orthonormal basis for \({{\mathscr {W}}}_n\). We fix such a basis uniquely by requiring that \(W_{j,n}=P_{j,n}\cdot e^{\frac{1}{2}nQ}\) where \(P_{j,n}\) is of exact degree j and has positive leading coefficient.
1.2.5 Auxiliary regions
In the sequel we write
where M is a fixed positive constant (depending only on Q). The \(\delta _n\)neighbourhood of a set E will be denoted
where \(D(a,r)=\{z\, ;\, za<r\}\) is the Euclidean disc with center a and radius r.
1.3 Asymptotic results for admissible potentials
In the following, Q denotes an admissible potential in the sense of Sect. 1.2.
1.3.1 Szegő type asymptotics for the reproducing kernel
We have the following result; the definitions of the various ingredients are given in the preceding subsection. (In particular \(N(U,\delta _n)\) denotes the \(\delta _n\)neighbourhood of the exterior set U, cf. (1.21).)
Theorem 1.3
Fix constants \(\eta \) and \(\beta \) with \(\eta >0\) and \(0<\beta <\frac{1}{4}\). Assuming that
we have the asymptotic formula
The Oconstant is uniform for the given set of z and w (depending only on the parameters \(\eta ,M\) and the potential Q).
Example
When \(Q=z^2\) we have \({{\mathscr {Q}}}=1\) and \({{\mathscr {H}}}=0\) while \(\phi (z)=z\). We thus recover the asymptotic formula in (1.5).
In the offdiagonal case when z, w are exactly on the boundary, we recognize several exact cocycles which may be canceled from the expression (1.23) without changing the statistical properties of the corresponding determinantal process. Recall that a cocycle is just a function of the form \(c_n(z,w)=g_n(z)/g_n(w)\) where \(g_n\) is a continuous and nonvanishing function.
Corollary 1.4
Suppose that \(z,w\in \Gamma \) and \(z\ne w\). Then
is a cocycle and
The formula (1.24) is related to a question studied by Forrester and Jancovici in the paper [42] on Coulomb gas ensembles at the edge of the droplet, in the special case of the elliptic Ginibre ensemble. The physical picture is that the screening cloud about a charge at the edge has a nonzero dipole moment, which gives rise to a slow decay of the correlation function. In [42] an argument on the physical level of rigor, based on Jancovici’s linear response theory, is given, and a formula for \(K_n(z,w)^2\) is predicted in the case when z, w are on the boundary ellipse and \(z\ne w\). This formula is consistent with (1.24) in the special case of the elliptic Ginibre ensemble.
In the recent work [4], the elliptic Ginibre ensemble is studied by using properties of the particular (Hermite) orthogonal polynomials which enter in that case. As a result, some more refined asymptotic results can be obtained in this case. A comparison is found in [4, Remark I.4] as well as in Sect. 1.4 below.
Remark
It is interesting to view the slow decay of chargecharge correlations in light of the fact that fluctuations near the boundary converge to a separate Gaussian field, which is independent from the one emerging in the bulk, see [11, 68] for the case of random normal matrices; details can be found in [10, Subsection 7.3]. The emergence of a separate boundary field makes it credible that a charge at the edge should correlate much stronger with other charges at the edge than with charges is the bulk, and our present results demonstrate that this expected behaviour is, in a broad sense, valid. (One should not read too much into the above analogy; after all, fluctuations converge in a weak, distributional sense, while our present results provide different, uniform estimates, for example for the connected 2point function \(K_n(z,w)^2\).)
We refer to Forrester’s recent survey article [39] as a source for many other kinds of fluctuation theorems. We may recall in particular that in settings of planar \(\beta \)ensembles, the two papers [22, 60] appeared almost simultaneously, suggesting two very different approaches to the question of proving Gaussian field convergence. (The case under study corresponds to \(\beta =2\) and was settled in [11, 68].)
1.3.2 Gaussian convergence of Berezin measures
Let \(K_n(z,w)\) be the reproducing kernel with respect to an arbitrary admissible potential Q.
Naturally, we define Berezin kernels and Berezin measures by
It is convenient to recall a few facts concerning these measures.

(1)
If z is a nondegenerate bulk point (in the sense that \(z\in {\text {Int}}S\) and \(\Delta Q(z)>0\)), then \(\mu _{n,z}\) converges to the Dirac point mass \(\delta _z\), whereas if \(z\in U\), then \(\mu _{n,z}\) converges to the harmonic measure \(\omega _z\) evaluated at z; the convergence holds in the weak sense of measures on \({{\mathbb {C}}}\). (See [10, Theorem 7.7.2].)

(2)
If z is a nondegenerate bulkpoint, then the convergence \(\mu _{n,z}\rightarrow \delta _z\) is Gaussian in the sense of heatkernel asymptotic: \(B_n(z,w)=n\Delta Q(z)\cdot e^{n\Delta Q(z)\,wz^2}\cdot (1+o(1)),\) where \(o(1)\rightarrow 0\) uniformly for (say) \(w\in D(z,\delta _n)\). (See e.g. [9].)

(3)
If \(z\in U\), then the weak convergence \(\mu _{n,z}\rightarrow \omega _z\) may be combined with an asymptotic result for the socalled rootfunction in [53, Theorem 1.4.1], indicating that the convergence must in a sense be “Gaussian”.
We shall now state a result giving a quantitative Gaussian approximation to \(\mu _{n,z}\) from which the convergence to harmonic measure will be directly manifest. For this purpose we express points w in some neighbourhood of \(\Gamma \) as
where \(p=p(w)\) is a point on \(\Gamma \), \(\mathtt{{n}}_1(p)\) is the unit normal to \(\Gamma \) pointing outwards from S, and \(\ell \) is a real parameter. (So \(\ell ={\text {dist}}(w,\Gamma )\) if \(\ell \) is close to 0.)
Given a point \(p\in \Gamma \) we also define a Gaussian probability measure \(\gamma _{p,n}\) on the real line by
For a given point \(z\in U\), we denote by \(\omega _z\) the harmonic measure of U evaluated at z and consider the measure \({\tilde{\mu }}_{n,z}\) given in the coordinate system (1.25) by
To be more explicit, we define the Poisson kernel \(P_z(p)\) as the density of \(\omega _z\) with respect to arclength dp on \(\Gamma \), i.e.,
Then
For this definition to be consistent, we fix a small neighbourhood of \(\Gamma \) and define \({\tilde{\mu }}_{n,z}\) by (1.28) in this neighbourhood and extend it by zero outside the neighbourhood. Then \({\tilde{\mu }}_{n,z}\) is a subprobability measure whose total mass quickly increases to 1 as \(n\rightarrow \infty \).
Theorem 1.5
Suppose that z is in the exterior component U. Then
where \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \) with uniform convergence when w is in the belt \(N(\Gamma ,\delta _n)=\{\ell \le \delta _n\}.\)
In other words, \(\mu _{n,z}=(1+o(1)){\tilde{\mu }}_{n,z}\) where \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \) in the sense of measures on \({{\mathbb {C}}}\) as well as in the uniform sense of densities on \(N(\Gamma ,\delta _n)\).
Remark
The last statement in Theorem 1.5 is automatic once the uniform convergence in (1.29) is shown. Indeed, let \(\epsilon >0\) be given. It is clear from the definition (1.28) that we can find M and \(n_0\) such that \({\tilde{\mu }}_{n,z}(N(\Gamma ,\delta _n))>1\epsilon \) when \(n\ge n_0\). Then \({\tilde{\mu }}_{n,z}({{\mathbb {C}}}\setminus N(\Gamma ,\delta _n))<\epsilon \) when \(n\ge n_0\). Thus \({\tilde{\mu }}_{n,z}_{{{\mathbb {C}}}\setminus N(\Gamma ,\delta _n)}\rightarrow 0\) as measures when \(n\rightarrow \infty \). By the uniform convergence in (1.29) we now see that \(\mu _{n,z}_{{{\mathbb {C}}}\setminus N(\Gamma ,\delta _n)}\rightarrow 0\), since \(\mu _{n,z}\) has unit total mass. Thus it suffices to prove the uniform convergence in (1.29); this is done in Sect. 4.
1.3.3 Main strategy: tailkernel approximation
An underpinning idea is that for points z and w in or close to the exterior set U, a good knowledge of the tail kernel
should suffice for deciding the leadingorder asymptotics of the full kernel \(K_n(z,w)\).
Note that \({\tilde{K}}_n(z,w)\) is just the reproducing kernel for the orthogonal complement \({\tilde{{{\mathscr {W}}}}}_n={{\mathscr {W}}}_n\ominus {{\mathscr {W}}}_{k,n}\) where \({{\mathscr {W}}}_{k,n}\subset {{\mathscr {W}}}_n\) is the subspace consisting of all \(W=P\cdot e^{\frac{1}{2} nQ}\) where P has degree at most \(k=\) “largest integer which is strictly less than \(n\theta _n\)”.
We shall deduce asymptotics for \({\tilde{K}}_n(z,w)\) using a technique based on summing by parts with the help of an approximation formula for \(W_{j,n}\) found in the paper [54].
When this is done, some fairly straightforward estimates for the lower degree terms (with \(j\le n\theta _n\)) are sufficient to show that the full kernel \(K_n(z,w)\) has similar asymptotic properties as does \({\tilde{K}}_n(z,w)\).
The practical execution of this strategy forms the bulk of this paper, cf. Sects. 3, 4, and 5.
1.4 The elliptic Ginibre ensemble
We now temporarily specialize to the elliptic Ginibre potential
where \(a,b>0\). It is convenient to assume that \(a<b\).
Remark
In the literature on the topic it is common to restrict to potentials depending on one single “nonHermiticity parameter” \(\tau \) with \(1<\tau <1\) and set the parameters in (1.31) to \(a=\frac{1}{1+\tau }\) and \(b= \frac{1}{1\tau }\), giving
However, other conventions are sometimes used, e.g. [62] takes \(a=1\tau \) and \(b=1+\tau \) while [7] fixes \(a=\frac{1}{2}\) and uses b as a (large) parameter.
It is easy to construct random samples with respect to the potential (1.31): start with two independent \(n\times n\) GUE matrices \(J_1\) and \(J_2\) and look at the random matrix
The eigenvalues \(\{z_j\}_1^n\) of \(X_n\) then correspond precisely to a random sample from the determinantal npoint process in potential Q; this is what was used to produce Fig. 3.
We now recast some wellknown facts about the elliptic Ginibre pointprocess; proofs and further details can be found in [3, 4, 7] and the references there.
In terms of the Hermite polynomials \(H_j(z)=(1)^je^{z^2}\frac{d^j}{dz^j}e^{z^2}\), the correlation kernel \(K_n(z,w)\) is given by
(This formula was used to plot Fig. 4.) Moreover, the droplet is the elliptic disc
which has its major semiaxis along the real line. The normalized conformal map \(\phi \) taking \(U=\hat{{{\mathbb {C}}}}\setminus S\) to \({{\mathbb {D}}}_e\) is the inverse Joukowsky map (wellknown from the theory of conformal mapping [65])
where \(\alpha = \tfrac{1}{2} ( \sqrt{\tfrac{2b}{a^2+ab}} + \sqrt{\tfrac{2a}{b^2+ab}} )\) and \(\beta = \tfrac{1}{2} ( \sqrt{\tfrac{2b}{a^2+ab}} \sqrt{\tfrac{2a}{b^2+ab}} )\). (Here we use the principal branch of the squareroot, so \(\phi (z)\sim z/\alpha \) as \(z\rightarrow \infty \) and \(\phi '(\infty )=1/\alpha \).)
Since the Laplacian \(\Delta Q\) is the constant \(\frac{1}{2}(a+b)\), we have \({{\mathscr {H}}}\equiv \frac{1}{2}\log (a+b)\). Inserting these data, our Theorem 1.3 (and using \({\text {Re}}{{\mathscr {Q}}}=Q\) on \({\partial }S\)) we obtain an effective approximation formula, which is consistent with the earlier predictions due to Forrester and Jancovici [42] as well as with more recent work due to Akemann, Duits and Molag [4]. We now comment on these works.
In the setting of Forrester and Jancovici, the key object is \(K_n(z,w)^2\) rather than the reproducing kernel \(K_n(z,w)\) itself. Forrester and Jancovici use linear response theory and asymptotics of Hermite polynomials to predict an asymptotic formula for \(K_n(z,w)^2\) in the offdiagonal case, when z, w belong to the boundary ellipse. With some effort, their formula can be shown to be consistent with Theorem 1.3 (and Corollary 1.4). Details can be found in the recent paper [4], see especially Remark I.4 for a comparison with our present work.
In the paper [4], the authors use different methods, relying on a contour integral representation of the kernel (1.32) and a saddle point analysis. Several refined results are derived there, notably [4, Theorem I.1], which among other things implies that the exterior type asymptotics for \(K_n(z,w)\) (from Theorem 1.3) persists in some fixed, nindependent neighbourhood of the boundary of the droplet (and away from the diagonal \(z=w\)). (When specialized to the Ginibre ensemble, this fact can of course be seen from Theorem 1.1 as well.) By contrast, Theorem 1.3 only guarantees asymptotics for \(K_n(z,w)\) when z, w belong to the shrinking neighbourhood \(N(U,\delta _n)\), of distance \(\delta _n\) from the boundary. Interestingly, the asymptotic formula [4, Theorem I.1] extends to the case when the points z, w stay away from the “motherbody”, i.e. the linesegment between the foci of the ellipse, and such that \(\phi (z)\overline{\phi (w)}1\ge \eta \) for some \(\eta >0\), again see [4, Remark I.4]. In particular, this provides information about the transition from exterior to bulktype asymptotics, in the elliptic Ginibre case.
1.5 Further results and related work
A good motivation for studying the reproducing kernel \(K_n(z,w)\) comes from random matrix theory, where it corresponds precisely to the “canonical correlation kernel”, e.g. [2, 12, 40, 64, 69]. If the external potential Q satisfies \(Q=+\infty \) on \({{\mathbb {C}}}\setminus {{\mathbb {R}}}\) we obtain Hermitian random matrix theory and Coulomb gas processes on \({{\mathbb {R}}}\), while if Q is admissible in our present sense, we obtain normal random matrix theory and planar Coulomb gas processes. Asymptotics for correlation kernels of normal random matrix ensembles has been the subject of many investigations, see for example [6, 12, 53, 54, 62] and the references there.
It is noteworthy that Forrester and Honner in the paper [41] study a different problem on edgecorrelations, between zeros of random polynomials \(p_n(z)=\sum _{j=0}^{n1} (j!)^{\frac{1}{2}} a_jz^j\) where the \(a_j\) are i.i.d. standard complex Gaussians. (The “edge” here is the circle \(z=\sqrt{n}\).)
Szegő’s paper [72] concerns zeros of partial sums \(S_n(z)=1+z+\cdots +\frac{z^n}{n!}\) of the Taylor series for \(e^z\).
It is not surprising that Szegő’s results should have a bearing for the Ginibre ensemble, since a factor \(S_{n1}(nz{\bar{w}})\) enters naturally in the formula (1.1). This has been used, for instance, in the papers [9, 50]. The Szegő curve (1.3) also enters in connection with the asymptotic analysis of various orthogonal polynomials, notably such which are associated with lemniscate ensembles, see [19, 20, 24, 63], cf. also Sect. 6.6 below. Szegő’s work can also be seen as a starting point for the theory of sections of power series of entire functions, cf. for instance [37, 76].
The sum \(S_{n1}(nz{\bar{w}})\) also has a close relationship to the upper incomplete gamma function \(\Gamma (a,z)=\int _z^\infty t^{a1}e^{t}\, dt\), via the identity (see [67, (Eq. 8.4.19)])
Thus we have the identity
Asymptotics for \(\Gamma (n,n\zeta )\) as \(n\rightarrow \infty \) in the case when \(\arg (\zeta 1)< \tfrac{3\pi }{4}\) can be deduced from Tricomi’s relation in [75, (Eq. 11)], see the NIST handbook [67, (Eq. 8.11.9)] as well as [66, (Eq. 2.2)] and the paper [46]. The formula is reproduced in (1.36) below.
Using the form in [67] we obtain readily that if \(\arg (\zeta 1)< \tfrac{3\pi }{4}\) then
where \(b_0(\zeta )=1\) and
Via Stirling’s formula (see Lemma 2.1 below) we can now conclude Theorem 1.1 in the case \(\arg (z{\bar{w}}1)<\tfrac{3\pi }{4}\). Conversely, we can use Theorem 1.1 to conclude the following generalized version of Tricomi’s expansion.
Corollary 1.6
The asymptotic expansion
holds for all \(\zeta \) in the exterior Szegő domain \(E_{\textrm{sz}}\). The domain \(E_{\textrm{sz}}\) is moreover the largest possible domain in which the expansion (1.36) holds.
Remark
The complete large n asymptotics of \(\Gamma (n,n\zeta )\) for \(\zeta \) in the complex plane may be deduced by using bulk asymptotics in Theorem 2.2 when \(\zeta \) is inside or on the Szegő curve, or errorfunction asymptotics when \(\zeta \) is very close to the critical point 1. We remark that a different kind of global asymptotics for the incomplete gamma function is given [66, 74]. In a way, our above results show that the asymptotics discussed in those sources can be simplified further, and in different ways, depending on whether \(\zeta \) is inside or outside of the Szegő curve.
As already indicated, we will make use of (and develop) the method of approximate fullplane orthogonal polynomials from the paper [54]. Such orthogonal polynomials are sometimes called Carleman polynomials [55]. In addition, we want to point to the paper [53], which studies the “root function”, essentially the Bergman space counterpart to the function
This is just the weighted polynomial squareroot of the Berezin kernel: \(B_n(z,w)=k_n(z,w)^2\). For z and w in appropriate regimes, an asymptotic expansion for \(k_n(z,w)\) can be deduced from [53, Theorem 1.4.1]. In Sect. 6.4 we shall use this expansion to deduce qualitative information concerning the structure of Berezin kernels.
A different (and very successful) approach in the theory of fullplane orthogonal polynomials is found in the paper [18], where strong asymptotics with respect to certain special types of potentials is deduced using RiemannHilbert techniques. In recent years, a number of other particular ensembles of intrinsic interest have turned out to be tractable by this method, see for instance the discussion in Sect. 6.6 below. In [58] it is noted that planar orthogonal polynomials can be characterized as the unique solution to a certain matrixvalued \(\bar{\partial }\)problem. In the recent papers [49, 52], related ideas are used to study fine asymptotics for orthogonal polynomials, leading to some additional insights besides the original approach in [54] (which uses foliation flows, as we do below).
In Sect. 6, our main results are viewed in relation to the loop equation. Some further results and a comparison with other relevant work is found there.
1.6 Plan of this paper
In Sect. 2 we consider the Ginibre ensemble and prove Theorem 1.1 and Corollary 1.2. In Sect. 3 we provide some necessary background for dealing with more general random normal matrix ensembles.
In Sect. 4, we state an approximation formula for \({\tilde{K}}_n(z,w)\) in (1.30) valid when z and w belong to \(N(U,\delta _n)\). This formula expresses \({\tilde{K}}_n(z,w)\) as a sum of certain weighted “quasipolynomials”, which have the advantage of being analytically more tractable than the actual orthogonal polynomials. Summing by parts in this formula we deduce Theorem 1.3 and Theorem 1.5.
In Sect. 5, we provide a selfcontained proof of the main approximation lemma used in Sect. 4. Our exposition is based on the method in [54], but is easier since (for example) we only require leading order asymptotics.
In Sect. 6 we view our main results in the context of the loop equation (or Ward’s identity). This leads to a hierarchy of identities relating the Berezin measures with various nontrivial (geometrically significant) objects.
1.7 Basic notation and terminology
Discs: \(D(a,r)=\{z\in {{\mathbb {C}}}\,;\,za<r\}\); \({{\mathbb {D}}}_e(r)=\{z>r\}\cup \{\infty \}\); \({{\mathbb {D}}}_e={{\mathbb {D}}}_e(1)\);
Neighbourhood of a set E: \(N(E,r)=E+D(0,r)\).
Differential operators: \({\partial }=\tfrac{1}{2}({\partial }_xi {\partial }_y)\), \(\bar{\partial }=\tfrac{1}{2}({\partial }_x+i {\partial }_y)\), \(\Delta ={\partial }\bar{\partial }\).
Area measure: \(dA=\tfrac{1}{\pi }\, dxdy\).
\(L^2\)scalar product and norm: \((f,g)=\int _{{\mathbb {C}}}f{\bar{g}}\, dA\); \(\Vert f\Vert =\sqrt{(f,f)}\).
Asymptotic relations: Given two sequences \(a_n\) and \(b_n\) of positive numbers we write: \(a_n\sim b_n\) if \(\lim _{n\rightarrow \infty }a_n/b_n= 1\); \(a_n\lesssim b_n\) if \(a_n/b_n\le C\) (C some constant); \(a_n\asymp b_n\) if \(a_n\lesssim b_n\) and \(b_n\lesssim a_n\).
2 Szegő’s Asymptotics and the Ginibre kernel
In this Section we prove Theorem 1.1 and Corollary 1.2 on asymptotics for the Ginibre kernel \(K_n(z,w)\) in the case when \(z{\bar{w}}\) belongs to the exterior Szegő domain \(E_{\textrm{sz}}\). In addition, we shall state and prove Theorem 2.2 on bulk type asymptotics.
2.1 Proof of Theorem 1.1
We start by writing the Ginibre kernel (1.1) in the form
where
A differentiation shows that
Following Szegő [72] we shall integrate in (2.2) along certain judiciously chosen paths.
The proof of the following lemma is straightforward from the usual Stirling series for \(\log n!\) (e.g. [1]).
Lemma 2.1
There are numbers \(b_k\) starting with \(b_0=1\) and \(b_1=\frac{1}{12}\) such that, for each \(k\ge 0\),
We now define a curve K and three regions \(\textrm{I},\textrm{II},\textrm{III}\) using the function \(u(\zeta )=\zeta \, e^{\,1\zeta }\), depicted in Fig. 5. The regions \(\textrm{I}\) (bounded) and \(\textrm{II}\) (unbounded) are defined to be the connected components of the set \(\{u(\zeta )<1\}\). (Note that \(\textrm{I}={\text {Int}}\gamma _{\textrm{sz}}\) is the domain interior to the Szegő curve (1.3).) We also define \(\textrm{III}:=\{u(\zeta )>1\}\).
Note that \(u(\zeta )\) has a critical point at \(\zeta =1\). We define the curve K to be the portion of the level curve \({\text {Im}}u(\zeta )=0\) which intersects the real axis at right angles at \(\zeta =1\). We assume that \(\zeta \ne 1\) and divide in two cases according to which \(\zeta \) is to the left or to the right of the curve K. (The case when \(\zeta \) is exactly on \(K\setminus \{1\}\) will be handled easily afterwards.)
First assume that \(\zeta \) is strictly to the right of K. (So \(\zeta \) is either in region \(\textrm{II}\) or in region \(\textrm{III}\) or on the common boundary of those regions.)
We integrate in (2.2) over the curve connecting \(\zeta \) to \(\infty \) in a way so that the argument of u(t) remains constant when t traces the path of integration. The path is chosen so that \({\text {Re}}t\rightarrow +\infty \) as \(t\rightarrow \infty \) along the curve; Fig. 6 illustrates the point. We find
A curve on which \(\arg u\) is constant is a steepest decent curve for \(\log u\) by the CauchyRiemann equations. This gives that u(t) strictly decreases from \(u(\zeta )\) to zero as t traces the curve from left to right. Thus we may unambiguously define an inverse function t(u) along the curve and obtain
From \(te^{1t}=u\) we obtain
so the last integral reduces to
Now write \(f(u)=(t(u)1)^{1}\) and consider the point
Then \(f(A)=(\zeta 1)^{1}\) and a (formal) repeated integration by parts gives
where as before u has constant argument along the path of integration, say \(u=e^{i\theta }x\) where \(0\le x\le A\).
Setting \({\tilde{f}}(x)=f(e^{i\theta }x)\) and \(M=\max \limits _{0\le x\le A}\{{\tilde{f}}^{(k+1)}(x)\}\) we obtain the estimate
Now \(f(u)=(t(u)1)^{1}\) gives
Inserting here \(u=u(\zeta )=A\) and using that \(t(u(\zeta ))=\zeta \) and
we obtain
By induction, one shows easily that the higher derivatives have the structure
where \(r_j(\zeta )\) is a rational function having a pole of order \(2j+1\) at \(\zeta =1\) and no other poles in \(\hat{{{\mathbb {C}}}}\).
On account of (2.5), (2.6), (2.7) and since \(f(A)=(\zeta 1)^{1}\) we have shown that
where \({\tilde{r}}_j(z)\) is a new rational function with pole of order 2j at \(\zeta =1\).
Recalling that \(\zeta =z{\bar{w}}\) and using (2.3) and Lemma 2.1,
where the expression in brackets is short for
and each \(\rho _j(\zeta )\) is a rational function with a pole of order 2j at \(\zeta =1\) and no other poles.
We have arrived at the expansion formula (1.4) in the case when \(\zeta =z{\bar{w}}\) is strictly to the right of the curve K.
Next we suppose that \(\zeta =z{\bar{w}}\) is strictly to the left of the curve K. (Thus \(\zeta \) is either in \(\textrm{I}\) or in \(\textrm{II}\) or on the common boundary of these domains.)
This time we can find a curve of constant argument of \(u(t)=te^{1t}\) connecting 0 with z, along which u(t) is strictly increasing. See Fig. 7.
We now integrate in (2.2) (using the fact that \(E_n(0)=1\)) to write
where
The path of integration is the curve of constant argument of u(t) indicated above.
As before, letting t(u) be the inverse function we find
By Stirling’s approximation (Lemma 2.1) and the asymptotic expansion (2.8),
where the expression in brackets is precisely the same as in (2.9).
Recalling that \(\zeta =z{\bar{w}}\) we obtain, as a consequence of (2.10) and (2.11) that
and hence, by (2.1),
The asymptotic formula in (2.12) has proven for all \(\zeta =z{\bar{w}}\) to the left of the curve K.
We next note that if \(\zeta =z{\bar{w}}\) is in the region \(\textrm{III}\), i.e., if \(z{\bar{w}}e^{1z{\bar{w}}}>1\), then the first term “1” inside the bracket in (2.12) is negligible, so in this case
as desired.
There remains to treat the case when \(\zeta =z{\bar{w}}\) happens to be precisely on the curve K and \(\zeta \ne 1\). In this case, we consider nearby points \(\zeta '\) which are either to the left or to the right of K and use a limiting procedure, as \(\zeta '\rightarrow \zeta \) to deduce that the asymptotic formula (1.4) is true in this case as well. (Intuitively, one can picture that for \(\zeta \in K\) we connect \(\zeta \) either to 0 or to \(+\infty \) by first following the curve K until we reach \(t=1\), and then continue along the real axis until we reach either 0 or \(+\infty \). This picture is however not entirely rigorous, since \(\frac{dt}{du}\) has a pole at at \(u=t=1\).)
Our proof of Theorem 1.1 is complete. q.e.d.
2.2 Bulk asymptotics for the Ginibre kernel
As a corollary of our above proof, we also obtain the following bulk type asymptotic expansion. (A related statement is found in [27, Proposition 2].)
Theorem 2.2
For \(z{\bar{w}}\in {{\mathbb {C}}}\setminus (E_{\textrm{sz}}\cup \{1\})\) we write \(\rho =z{\bar{w}}e^{1z{\bar{w}}}\). Then \(\rho \le 1\) and we have the bulkasymptotic formula
where the implied Oconstant is uniform for \(z{\bar{w}}\) in the complement of any neighbourhood of 1.
Proof
The asymptotic formula in (2.12) applies since \(\zeta =z{\bar{w}}\) is on the left of the curve K under the assumptions in Theorem 2.2. Moreover the second term inside the bracket in (2.12) is \(O(n^{\frac{1}{2}}\zeta e^{1\zeta }^n\zeta 1^{1})\). \(\square \)
Remark
It follows that if \(z{\bar{w}}\in {{\mathbb {C}}}\setminus (E_{\textrm{sz}}\cup \{1\})\), then the Berezin kernel \(B_n(z,w)\) satisfies the heatkernel asymptotic \(B_n(z,w)=ne^{nzw^2}\cdot (1+o(1))\). This has been wellknown when the points z and w are close enough to the diagonal \(z=w\) and in the interior of the droplet, cf. [6, 9, 12]. The main point in Theorem 2.2 is that we obtain the precise domain of “bulk asymptoticity”.
2.3 Proof of Corollary 1.2
Fix a (finite) point z in the exterior disc \({{\mathbb {D}}}_e\), and consider the Berezin measure \(d\mu _{n,z}(w)=B_n(z,w)\, dA(w).\) We aim to prove that \(\mu _{n,z}\) converges to the harmonic measure \(\omega _z\) in a Gaussian way.
For this purpose, we fix a sequence \((c_n)\) of positive numbers with \(nc_n^2\rightarrow \infty \) and \(nc_n^3\rightarrow 0\) as \(n\rightarrow \infty \); we can without loss of generality assume that \(c_n<1\) for all n. We then consider points w in the belt \(N({{\mathbb {T}}},c_n)\), represented in the form
A computation shows that
Let \(\hat{\mu }_{n,z}(\theta ,t)=\mu _{n,z}\circ f_n(\theta ,t)\) be the pullback of \(\mu _{n,z}\) by \(f_n\). Also fix \(\theta \in {{\mathbb {T}}}\) and consider the radial crosssection
Since \(d\hat{\mu }_{n,z}(\theta ,t)={\varrho }_{n,\theta }(t)\, d\theta dt\), it suffices to study asymptotics of the function \({\varrho }_{n,\theta }(t)\).
Fixing t we now define an ndependent point \(w=f_n(\theta ,t)\), as in (2.14). Then by Theorem 1.1
After some simplification using that \(f_n(\theta ,t)=e^{i\theta }(1+o(1))\) and \(f_n(\theta ,t)=1+o(1)\) as \(n\rightarrow \infty \), we find that
where \(P_z(\theta )=\frac{1}{2\pi }\frac{z^21}{1z e^{i\theta }^2}\) is the Poisson kernel.
We next observe that
so (since \(nc_n^3\rightarrow 0\)) we obtain, with \(\gamma (t)=\frac{1}{\sqrt{2\pi }}e^{\frac{1}{2} t^2}\),
where the last equality follows by straightforward simplification.
Finally setting \(\ell =t/(2\sqrt{n})\) it follows that the Berezin measure \(\mu _{n,z}\) has the uniform asymptotic
where \(\gamma _n(\ell )=2\sqrt{n}\frac{1}{\sqrt{2\pi }}e^{2n\ell ^2}\, d\ell \). q.e.d.
3 Potential Theoretic Preliminaries
This section begins by recalling how boundary regularity follows from Sakai’s main result in [70]. After that we recast some useful facts pertaining to Laplacian growth and obstacle problems. Finally we will state and prove a number of estimates for weighted polynomials, which will come in handy when approximating the reproducing kernel \(K_n(z,w)\) by its tail in the next section.
3.1 Sakai’s theorem on boundary regularity
Let Q be an admissible potential.
As always we denote by S the droplet and U the component of \(\hat{{{\mathbb {C}}}}\setminus S\) containing infinity.
We also write \(\Gamma ={\partial }U\) and \(\chi :{{\mathbb {D}}}_e\rightarrow U\) for the conformal mapping that satisfies \(\chi (\infty )=\infty \) and \(\chi '(\infty )>0\).
Lemma 3.1
Let p be an arbitrary point on \(\Gamma \). There exists a neighbourhood N of p and a “local Schwarz function”, i.e., a holomorphic function \({{\mathscr {S}}}(z)\) on \(N\setminus S\), continuous up to \(N\cap \Gamma \) and satisfying \({{\mathscr {S}}}(z)={\bar{z}}\) there.
Proof
Without loss of generality set \(p=0\).
Choosing the neighbourhood N sufficiently small we can write \(Q(z)=\sum _{j,k=0}^\infty a_{j,k} z^j{\bar{z}}^k\) with convergence for all z in N. By polarization we define \(H(z,w)=\sum _{j,k=0}^\infty a_{j,k} z^jw^k.\)
Next define a Lipschitzian function G(z, w) in \(N\times N\) by
Here \({\check{Q}}\) is the obstacle function, defined in Subsection 1.2.
In \(N\setminus S\), the function \({\partial }{\check{Q}}\) is holomorphic and we further have that \({\partial }{\check{Q}}={\partial }Q\) on \(N\cap ({\partial }S)\). Hence G(z, w) is a holomorphic function of z for \(z\in N\setminus S\), \(G(0,0)=0\) and \({\partial }_w G(z,w)_{(0,0)}=\Delta Q(0)>0\). Moreover, the identity \(Q(z)=H(z,{\bar{z}})\) shows that
so \(G(z,{\bar{z}})=0\) for all \(z\in N\cap S\), and, in particular, for all \(z\in N\cap ({\partial }S)\).
By the implicit function theorem (in its version for Lipschitz functions [34]) we may, by diminishing N if necessary, find a unique Lipschitzian solution \({{\mathscr {S}}}(z)\) to the equation \(G(z,{{\mathscr {S}}}(z))=0\), \(z\in N\).
Then for \(z\in N\setminus S\) we obtain \(0=\bar{\partial }_z G(z,{{\mathscr {S}}}(z))={\partial }_2 G(z,{{\mathscr {S}}}(z))\cdot \bar{\partial }{{\mathscr {S}}}(z)\), proving that \({{\mathscr {S}}}(z)\) is holomorphic in \(N\setminus S\).
We also find that \(G(z,{\bar{z}})=G(z,{{\mathscr {S}}}(z))=0\) for \(z\in N\cap ({\partial }S)\), so \({{\mathscr {S}}}(z)={\bar{z}}\) at such points. \(\square \)
Theorem 3.2
The conformal map \(\chi :{{\mathbb {D}}}_e\rightarrow U\) extends analytically across \({{\mathbb {T}}}\) to an analytic function on a neighbourhood of the closure of \({{\mathbb {D}}}_e\). As a consequence \(\Gamma =\chi ({{\mathbb {T}}})\) is a finite union of real analytic arcs and possibly finitely many singular points, which are either cusps (corresponding to points \(p=\chi (z)\) with \(z\in {{\mathbb {T}}}\) and \(\chi '(z)=0\)) or double points (\(p=\chi (z_1)=\chi (z_2)\) where \(z_1,z_2\in {{\mathbb {T}}}\) and \(z_1\ne z_2\)).
The proof is immediate from Sakai’s regularity theorem in [70], since \(\Gamma \) is a continuum and since a local Schwarz function for U exists near each point of \(\Gamma \) by Lemma 3.1.
3.2 Laplacian growth and Riemann maps
For fixed \(\tau \in (0,1]\) we let \({\check{Q}}_\tau \) be the obstacle function which grows like
By this we mean that \({\check{Q}}_\tau (z)\) is the supremum of s(z) where s runs through the class \({\textrm{SH}}_\tau (Q)\) of subharmonic functions s on \({{\mathbb {C}}}\) which satisfy \(s\le Q\) on \({{\mathbb {C}}}\) and \(s(w)\le 2\tau \log w+O(1)\) as \(w\rightarrow \infty \).
Similar as for the case \(\tau =1\), the function \({\check{Q}}_\tau \) is \(C^{1,1}\)smooth on \({{\mathbb {C}}}\) and harmonic on \({{\mathbb {C}}}\setminus S_\tau \) where \(S_\tau =S[Q/\tau ]\) is the droplet in potential \(Q/\tau \), while \(Q={\check{Q}}_\tau \) on \(S_\tau \). (See [61, 69].)
Clearly the droplets \(S_\tau \) increase with \(\tau \); the evolution is known as Laplacian growth, cf. [48, 54, 61, 77].
Recall that \(d\sigma =\Delta Q\cdot {\textbf{1}}_S\, dA\) denotes the equilibrium measure in external potential Q. It is easy to see that \(\sigma (S_\tau )=\tau ,\) and that the restricted measure \(\sigma _\tau \) defined by
minimizes the weighted energy \(I_Q[\mu ]\) in (1.15) among all compactly supported Borel measures \(\mu \) of total mass \(\mu ({{\mathbb {C}}})=\tau \). We refer to \(\sigma _\tau \) as the equilibrium measure of mass \(\tau \).
Write \(U_\tau \) for the component of \(\hat{{{\mathbb {C}}}}\setminus S_\tau \) containing \(\infty \) and \(\Gamma _\tau ={\partial }U_\tau \) for the outer boundary of \(S_\tau \).
By hypothesis, \(\Gamma =\Gamma _1\) is everywhere regular (realanalytic). From this and basic facts about Laplacian growth [48, 54] we conclude that there are numbers \(\tau _0<1\) and \(\epsilon >0\) such that \(\Gamma _\tau \) is everywhere regular whenever \(\tau _0\epsilon \le \tau \le 1\). Indeed \(\tau _0\) and \(\epsilon \) can be chosen so that each potential \(Q/\tau \) with \(\tau _0\epsilon \le \tau \le 1\) is admissible in the sense of Sect. 1.2.
We denote by \(\phi _\tau :U_\tau \rightarrow {{\mathbb {D}}}_e\) the conformal mapping normalized by \(\phi _\tau (\infty )=\infty \) and \(\phi _\tau '(\infty )>0\). We also write \(\mathtt{{n}}_\tau :\Gamma _\tau \rightarrow {{\mathbb {T}}}\) for the unit normal on \(\Gamma _\tau \) pointing out of \(S_\tau \).
Lemma 3.3
(“Rate of propagation of \(\Gamma _\tau \).”). The boundary \(\Gamma _\tau \) moves in the direction of \(\mathtt{{n}}_\tau \) with local speed \(\phi _\tau '/2\Delta Q\) in the following precise sense.
Pick two numbers \(\tau ',\tau \) in the interval \([\tau _0\epsilon ,1]\). Fix \(z\in \Gamma _{\tau '}\) and let p be the point in \(\Gamma _{\tau }\) which is closest to z. Then
and
where the Oconstants are uniform in z. In particular there are constants \(0<c_1\le c_2\) such that
For a proof we refer to [54, Lemma 2.3.1]. (Cf. [14, Lemma 5.2].)
For given \(\tau \) with \(\tau _0\le \tau \le 1\) we denote
Modifying \(\tau _0<1\) and \(\epsilon >0\) if necessary, we may assume that \(V_\tau \) is welldefined and harmonic on \({{\mathbb {C}}}\setminus K\) where K is a compact subset of \({\text {Int}}\Gamma _{\tau _0\epsilon }\). The set K can be chosen depending only on \(\tau _0\) and \(\epsilon \) and not on the particular \(\tau \) with \(\tau _0\le \tau \le 1\).
We shall frequently use the following identity:
where \({{\mathscr {Q}}}_\tau \) is the unique holomorphic function on \(U_\tau \) with \({\text {Re}}{{\mathscr {Q}}}_\tau =Q\) on \(\Gamma _\tau \) and \({\text {Im}}{{\mathscr {Q}}}_\tau (\infty )=0\). (This follows since the left and right sides agree on \(\Gamma _\tau \) and have the same order of growth at infinity.)
We turn to a few basic estimates for the function \(QV_\tau \), which we may call “\(\tau \)ridge”.
Lemma 3.4
Suppose that \(\tau _0\epsilon \le \tau \le 1\) and let p be a point on \(\Gamma _\tau \). Then for \(\ell \in {{\mathbb {R}}}\),
where the Oconstant can be chosen independent of the point \(p\in \Gamma _\tau \).
Proof
Using that \(Q={\check{Q}}_\tau \) on \(S_\tau \) and that \({\check{Q}}_\tau \) is \(C^{1,1}\)smooth, we find that \(\tfrac{{\partial }^2}{{\partial }n^2}(QV_\tau )(p)= (\tfrac{{\partial }^2}{{\partial }n^2}+\tfrac{{\partial }^2}{d s^2})(QV_\tau )(p)=4\Delta Q(p)\), where \(\tfrac{{\partial }}{{\partial }n}\) and \(\tfrac{{\partial }}{{\partial }s}\) denote differentiation in the normal and tangential directions, respectively. The result now follows from Taylor’s formula. \(\square \)
Our next lemma is immediate from Lemma 3.4 when z is close to \(\Gamma _\tau \) and follows easily from our standing assumptions on Q when z is further away (cf. the proof of [5, Lemma 2.1], for example).
Lemma 3.5
Suppose that \(\tau _0\epsilon \le \tau \le 1\) and that z is in the complement \({{\mathbb {C}}}\setminus K\), where \(K\subset {\text {Int}}\Gamma _{\tau _0\epsilon }\) is defined above. Then with \(\delta _\tau (z)={\text {dist}}(z,\Gamma _\tau )\) there is a number \(c>0\) such that
Combining Lemma 3.3 with Lemma 3.4, we now obtain the following useful result.
Lemma 3.6
Suppose that \(\tau ,\tau '\) are in the interval \([\tau _0{\varepsilon },1]\). For a given point \(z\in \Gamma _{\tau '}\) let \(p\in \Gamma _\tau \) be the point closest to z. Then
In particular, if \(\tau _0<1\) and \(\epsilon >0\) are chosen close enough to 1 and 0 respectively, then there are constants \(c_1\) and \(c_2\) independent of \(\tau \), \(\tau '\), z such that
Before closing this section, it is convenient to prove a few facts about weighted polynomials.
3.3 Pointwise estimates for weighted orthogonal polynomials
We now collect a number of estimates whose main purpose is to ensure a desired tailkernel approximation in the next section. To this end, the main fact to be applied is Lemma 3.10.
We start by proving the following pointwise\(L^2\) estimate, following a slight variation on a technique which is wellknown in the literature.
Lemma 3.7
Let \(W=P\cdot e^{\frac{1}{2} nQ}\) be a weighted polynomial where \(j=\deg P\le n\). Put \(\tau (j)=j/n\) and suppose \(\tau (j)\le \tau \) where \(\tau \) satisfies \(0<\tau \le 1\). There is then a constant C depending only on Q such that for all \(z\in {{\mathbb {C}}}\),
Proof
Let \(M_\tau \) be the maximum of W over \(S_\tau \). We shall first prove that
To this end we may assume that \(M_\tau = 1\). Consider the function
which is subharmonic on \({{\mathbb {C}}}\) and satisfies \(s\le Q\) on \(\Gamma _\tau \). Moreover, \(s(w)\le 2\tau \log w+O(1)\) as \(w\rightarrow \infty \). Hence by the strong maximum principle we have \(s\le {\check{Q}}_\tau \) on \({{\mathbb {C}}}\), proving (3.6).
We next observe that there is a constant C independent of \(\tau \) such that
Indeed, (3.7) follows from a standard pointwise\(L^2\) estimate, see for example [5, Lemma 2.4].
Combining (3.6) and (3.7) we finish the proof of the lemma. \(\square \)
We shall need to compare obstacle functions \({\check{Q}}_\tau (z)\) for different choices of parameter \(\tau \). It is convenient to note the following two lemmas.
Lemma 3.8
Suppose that \(\tau _0\le \tau \le \tau '\le 1\). Then there is a constant \(c>0\) depending only on \(\tau _0\) and Q such that
Proof
Write
Then H is harmonic on U (including infinity) and has boundary values \(H(z)=(QV_\tau )(z)\) for \(z\in \Gamma _{\tau '}\).
For a given \(z\in \Gamma _{\tau '}\) we let \(p\in \Gamma _\tau \) be the closest point and write \(z=p+\ell \cdot \mathtt{{n}}_\tau (p)\). Then \(\ell \asymp \tau '\tau \) by Lemma 3.3, and by Lemma 3.4\(H(z)=2\Delta Q(p)\cdot \ell ^2+O(\ell ^3).\)
Increasing \(\tau _0<1\) a little if necessary, we obtain \(H\ge c(\tau '\tau )^2\) everywhere on \(\Gamma _{\tau '}\) where \(c>0\) is a constant depending on \(\tau _0\) and Q. By the maximum principle, the inequality \(H\ge c(\tau '\tau )^2\) persists on \(U_{\tau '}\). \(\square \)
Lemma 3.9
Let \(W=P\cdot e^{\frac{1}{2} nQ}\) be a weighted polynomial where \(j=\deg P\le n\) with \(\Vert W\Vert =1\). Suppose also that \(\tau (j)\le \tau \) where \(\tau _0\le \tau \le \tau '\le 1\). Then there are constants C and \(c>0\) such that
Proof
Combining Lemma 3.7 with Lemma 3.8 we find that for all \(z\in {\text {cl}}U_{\tau '}\)
\(\square \)
Finally, we arrive at following estimate, which will be used to discard lower order terms in the tailkernel approximation in the succeeding section.
Lemma 3.10
Let
Suppose that \(\tau (j)\le \theta _n\) and let \(W_{j,n}(z)\) be the j:th weighted orthonormal polynomial in the subspace \({{\mathscr {W}}}_n\subset L^2({{\mathbb {C}}})\). There are then constants C and \(c>0\) depending only on Q such that
Proof
We may assume that \(\tau (j)\ge \tau _0\) where \(\tau _0<1\) is as close to 1 as we please. We apply Lemma 3.9 with
where \(C>0\) is chosen so that \(N(U,\delta _n)\subset U_{\tau '}\). It is possible to find such a C by Lemma 3.3.
Applying Lemma 3.9, we find that (with a new C)
Since \({\check{Q}}_{\tau '}\le {\check{Q}}\) and
we finish the proof by choosing \(c>0\) somewhat smaller. \(\square \)
4 Kernel Asymptotics: Proofs of the Main Results
In this section, we prove Theorem 1.3 on asymptotics for reproducing kernels, and Theorem 1.5 on Gaussian convergence of Berezin measures.
Throughout the section, we fix an external potential Q obeying the standing assumptions in Sect. 1.2.
4.1 The tail kernel
Consider the tail kernel
where \(W_{j,n}=P_{j,n}\cdot e^{\frac{1}{2}nQ}\), is the j:th weighted orthogonal polynomial, i.e., \(P_{j,n}\) has degree j and positive leading coefficient. The numbers \(\theta _n\) and \(\delta _n\) are defined by
where M is fixed (depending only on Q).
The following approximation lemma is our main tool; we remind once and for all that the symbol U denotes the component of the complement of the droplet S which contains \(\infty \).
Lemma 4.1
(“Main approximation lemma”). Suppose that
and let \(\beta \) be any fixed number with \(0<\beta <\tfrac{1}{4}\). Then with \(\tau (j)=\frac{j}{n}\) we have
Throughout this section, we will accept the lemma; a relatively short derivation, based on the method in [54], is given in Sect. 5.
We now turn to the proofs of our main results (Theorems 1.3 and 1.5).
Towards this end (using notation such as \({{\mathscr {Q}}}={{\mathscr {Q}}}_1\) and \(\phi =\phi _1\)) we rewrite (4.3) as
where we used the notation
with
Our main task at hand is to estimate the sum \({\tilde{S}}_n(z,w)\).
Remark
In going from (4.3) to (4.4) we used the facts that \({{\mathscr {H}}}_\tau (z)={{\mathscr {H}}}(z)+O(1\tau )\) and \(\phi '_\tau (z)=\phi '(z)+O(1\tau )\) as \(n\rightarrow \infty \) where the Oconstants are uniform for z in \(U_\tau \) and (say) \(n\theta _n\le \tau \le n\). This follows by an application of the maximum principle, using that the functions are holomorphic on \(\hat{{{\mathbb {C}}}}\setminus K\) and that relevant estimates are clear on the boundary curve \(\Gamma _\tau \).
We have the following main lemma, in which we fix a small number \(\eta >0\).
Lemma 4.2
Suppose that \(z,w\in N(U,\delta _n)\) and that \(\phi (z)\overline{\phi (w)}1\ge \eta \). Then there is a positive constant \(N\) such that
The constant N as well as the Oconstant can be chosen depending only on the parameters \(\eta \) and M, and on the potential Q.
Taken together with (4.4), the lemma gives a convenient approximation formula for the tail \({\tilde{K}}_n(z,w)\). We shall later find that the full kernel \(K_n(z,w)\) obeys the same asymptotic to a negligible error, for the set of z and w in question.
We first turn to our proof of Lemma 4.2 in the following two subsections. After that, the proof of Theorem 1.3 follows in Sect. 4.4.
4.2 Preparation for the proof of Lemma 4.2
For \(\tau \) close to 1 we introduce the following holomorphic function on \(\hat{{{\mathbb {C}}}}\setminus K\),
Notice that \(F_{\tau }(\infty )>0\) and that (4.6) can be written
For the purpose of estimating \({\tilde{S}}_n(z,w)\) we write
and
We also denote
the integer part of \(n\theta _n\).
Applying summation by parts, we write
where
The proof of the following lemma is immediate from (4.8).
Lemma 4.3
For all \(z,w\in {{\mathbb {C}}}\setminus K\),
We shall find below that \(a_{n1}\rightarrow 1\) and \(a_m\rightarrow 0\) quickly as \(n\rightarrow \infty \). Once this is done there remains to show that the penultimate term in the right hand side is negligible in comparison with the first one. This latter point is where our main efforts will be deployed.
4.3 Proof of Lemma 4.2
Throughout this subsection it is assumed that z and w belong to \(N(U,\delta _n)\) and that \(\phi (z)\overline{\phi (w)}1\ge \eta \), and we write \(\tau (j)=\frac{j}{n}\).
We begin with the following lemma.
Lemma 4.4
Let h(z) be the unique holomorphic function in a neighbourhood of \({\overline{U}}\) which satisfies the boundary condition
and the normalization \({\text {Im}}h(\infty )=0.\)
Then for all z, w in a neighbourhood of \({\overline{U}}\) and all j such that \(\tau _0\le \tau (j)\le 1\) we have as \(n\rightarrow \infty \)
where \(b_3(z)\) is a holomorphic function in a neighbourhood of \({\overline{U}}\).
Before proving the lemma, we note that the harmonic function \({\text {Re}}h(z)\) defined by the boundary condition (4.10) is strictly negative in a neighbourhood of \({\overline{U}}\) by the maximum principle.
Hence Lemma 4.4 implies the following result.
Corollary 4.5
By slightly increasing the compact set \(K \subset {{\mathbb {C}}}\setminus {\overline{U}}\) if necessary, we can ensure that for all \(z,w\in {{\mathbb {C}}}\setminus K\),
and there is a constant \(s>0\) such that (with \(m=n\theta _n\))
Moreover, s and the implied constants can be chosen uniformly for the given set of z and w.
Proof of Lemma 4.4
For \(z\in \hat{{{\mathbb {C}}}}\setminus K\) and real \(\tau \) near 1 we consider the function
where we use the principal determination of the logarithm, i.e., \({\text {Im}}P(\tau ,\infty )=0\). It is clear that
We now consider the Taylor expansion in \(\tau \), about \(\tau =1\),
where
is holomorphic in \(\hat{{{\mathbb {C}}}}\setminus K\) and \({\text {Im}}b_k(\infty )=0\).
Now using that
we conclude that
which we write as
If \(z\in \Gamma \), this reduces to
whence by the asymptotics in Lemma 3.6, we have as \(\tau \rightarrow 1\),
Comparing with (4.12) we infer that the holomorphic functions \(b_1\) and \(b_2\) on \(\hat{{{\mathbb {C}}}}\setminus K\) satisfy \({\text {Re}}b_1=0\) on \(\Gamma \) and
The normalization at infinity determines \(b_1=0\) and \(b_2=h\) uniquely, where h(z) is the function in the statement of the lemma.
To finish the proof, it suffices to observe that
and refer to (4.15). \(\square \)
At this point, it is convenient to switch notation and write
where then \(1\le k\le \sqrt{n}\log n\). We will denote
and assume that this is an integer. We will also write
The following lemma is a direct consequence of Lemma 4.4.
Lemma 4.6
For \(n\mu \le j\le n1\) we have the asymptotic (as \(n\rightarrow \infty \))
Proof
This is immediate on writing
noting that \(n({\varepsilon }_{k1}^2{\varepsilon }_{k}^2)=2{\varepsilon }_k+\frac{1}{n}\) and inserting the asymptotics in Lemma 4.4; details are left for the reader. \(\square \)
We are now ready to give our proof of Lemma 4.2.
Proof of Lemma 4.2
Pick \(z,w\in N(U,\delta _n)\) and write
where \(r>0\) and \(\vartheta \le \pi \).
We are assuming that \(re^{i\vartheta }1\ge \eta >0\). By continuity of the reflection in \({{\mathbb {T}}}\): \(re^{i\vartheta }\mapsto r^{1} e^{i\vartheta }\), there is also a constant \(\eta _0=\eta _0(\eta )>0\) such that
By Lemma 3.3 we have in addition that
for some constant C depending only on M and Q.
We now consider the sum
In view of Lemma 4.3 and Corollary 4.5, we shall be done when we can prove the bound
with some constant N.
Using Lemma 4.6, it is seen that
Here A, B, c are certain complex numbers depending on z and w; the important fact is that
To analyze the right hand side in (4.21), we set
and introduce the notation
From (4.18) we have the lower bound
Also, since \(r\ge 1C\delta _n\) we have
We now show that \(\sigma _{n,1}\) and \(\sigma _{n,2}\) are negligible as \(n\rightarrow \infty \).
To treat the case of \(\sigma _{n,1}\) we write
so that
Using (4.22) and (4.23) we see that the partial sums
obey the estimate \(B_k^{(1)}\lesssim e^{C\delta _n k}.\) In particular we have that
Let us write
Since \(a_\mu ^{(1)}\lesssim e^{\alpha \log ^2 n}\), a summation by parts gives
with any s satisfying \(0<s<\alpha \).
Next observe that
whence
Making use of a Riemann sum and the substitution \(t=\sqrt{n}{\varepsilon }\), we get
where we put \({\tilde{C}}=CM/2\).
In the case when r is “large” in the sense that \(r\ge r_0>1\), we can do better. Indeed since \({\text {Re}}d=\log r\), the partial sums \(B_k^{(1)}\) obey the bound \(B_k^{(1)}\lesssim r^{1}\) where the implied constant depends on \(r_0\). The method of estimation above thus gives
The term \(\sigma _{n,2}\) can be handled similarly: we introduce the notation
One deduces without difficulty that
A straightforward adaptation of our above estimates for \(\sigma _{n,1}\) now leads to
Moreover, in the case when \(r\ge r_0>1\) we obtain the improved estimate \(\sigma _{n,2}\lesssim 1/(r\sqrt{n})\).
The remaining terms in the right hand side of (4.21) will be estimated in a more straightforward manner, by taking the absolute values inside the corresponding sums.
Keeping the notation \(\alpha ={\text {Re}}c>0\) we thus consider the following four terms:
By a Riemann sum approximation and the estimate (4.19) we find
Since (for \(0\le \nu \le 3\))
we conclude that
It is also easy to verify that for \(r\ge r_0>1\) we have \({\tilde{\sigma }}_{n,\nu }\lesssim 1/(r\sqrt{n})\). (For example, one can sum by parts as above, using that the summation index k starts at 2.)
All in all, by virtue of the relation (4.21), we conclude the estimate (with a new C)
while if \(r\ge r_0>1\),
Combining these estimates, we find in all cases that \(\sigma _n\lesssim \frac{(\log n)^{C^2/\alpha ^2}}{\sqrt{n}}\, r^n\). We have reached the desired bound (4.20) with \(N=C^2/\alpha ^2\), and our proof of Lemma 4.2 is complete. \(\square \)
4.4 Proof of Theorem 1.3
In what follows we consider two arbitrary points \(z,w\in N(U,\delta _n)\) such that \(\phi (z)\overline{\phi (w)}1\ge \eta \).
Consider the full reproducing kernel
In view of Lemma 4.2 it suffices to prove that \(K_n(z,w)\) is, in a suitable sense, “close” to the tail kernel \({\tilde{K}}_n(z,w)\).
To prove this we first note that Lemma 4.2 implies that the size of the tailkernel is
To estimate lower order terms, corresponding to j with \(\tau (j)\le \theta _n\), we recall Lemma 3.10 that there is a number \(c'>0\) such that for all \(z\in N(U,\delta _n)\)
Using a similar estimate for \(W_{j,n}(w)\) and picking any \(c>0\) with \(c<c'\), we conclude the estimate
Since \({\check{Q}}=V\) on U, we obtain from (4.27) and (4.29) that \(K_n(z,w)= {\tilde{K}}_n(z,w)\cdot (1+O(e^{\frac{c}{2}\log ^2 n}))\) in the case when both z and w are in \({\overline{U}}\). However, since z and w are allowed to vary in the \(\delta _n\)neighbourhood, we require a slight extra argument.
We shall use the following simple lemma, which also appears implicitly in the proof of [14, Lemma 6.6].
Lemma 4.7
There is a constant C such that for all \(z\in N(U,\delta _n)\),
Proof
Since \({\check{Q}}=V\) on U, we can assume that \(z\in S\). Then \({\check{Q}}(z)=Q(z)\). Let \(p\in \Gamma \) be the closest point and write \(z=p+\ell \mathtt{{n}}_1(p)\) where \(\ell \lesssim \delta _n\) by Lemma 3.3. The Taylor expansion in Lemma 3.4 now shows that \((QV)(z)=2\Delta Q(p)\ell ^2+O(\ell ^3)\), finishing the proof of the claim. \(\square \)
Combining (4.29) with (4.30) we conclude that if \(z,w\in N(U,\delta _n)\) then
for a suitable positive constant \(c'\). Fix \(c''\) with \(0<c''<c\) and then pick a new \(c>0\) with \(c<c''\). Comparing with (4.27), we obtain
We have shown that
By Lemma 4.2, we know that the tail kernel \({\tilde{K}}_n(z,w)\) has the desired asymptotic when the points z, w belong to \(N(U,\delta _n)\) and \(\phi (z)\overline{\phi (w)}1\ge \eta \). Thus by (4.31) we find that \(K_n(z,w)\) obeys the same asymptotic, finishing our proof of Theorem 1.3. q.e.d.
4.5 Proof of Theorem 1.5
Fix a point \(z\in U\) and recall that
We express points w in \({{\mathbb {C}}}\) (in some fixed neighbourhood of \(\Gamma \)) as \(w=p+\ell \cdot \mathtt{{n}}_1(p)\) where \(p=p(w)\) is a point on \(\Gamma \), \(\mathtt{{n}}_1(p)\) is the unit normal to \(\Gamma \) pointing outwards from S and \(\ell \) is a real parameter.
Given a point \(p\in \Gamma \) we also recall the Gaussian probability measure \(\gamma _{p,n}\) on the real line,
Denote by \(\omega _z=\omega _{z,U}\) the harmonic measure of U evaluated at z and consider the measure \(d{\tilde{\mu }}_{n,z}=d\omega _z(p)\, d\gamma _{n,p}(\ell )\); writing \(d\omega _z(p)=P_z(p)\, dp\), we have
By Theorem 1.3 we have, for fixed \(z\in U\) and any \(w\in N(U,\delta _n)\),
Recalling that \(\phi (w)^{2n}e^{n{\text {Re}}{{\mathscr {Q}}}(w)}=e^{nV(w)}\), we obtain
We next recall that (by Lemma 3.5), the factor \(e^{n(QV)(w)}\) is negligible when \({\text {dist}}(w,\Gamma )\ge \delta _n\), so we can focus on the asymptotics of (4.33) in the \(\delta _n\)neighbourhood \(N(\Gamma ,\delta _n)\).
Near the curve \(\Gamma \), Lemma 3.4 gives
where Oconstant is independent of the point \(p\in \Gamma \).
Using (4.34) and (4.33) we infer that, when \(\ell \le \delta _n\),
We now change variables from \((p,\ell )\in \Gamma \times {{\mathbb {R}}}\) to \((\theta ,t)\in {{\mathbb {T}}}\times {{\mathbb {R}}}\) by the inverse of the mapping
In these coordinates, (4.35) becomes
An easy computation shows that
whence the pullback measure \(\mu _{n,z}\circ f_n\) satisfies
(The convergence \(o(1)\rightarrow 0\) holds in the uniform sense of densities on the sets where \(t\le 2M\sqrt{\log \log n}\).)
At this point we notice that the measure
is precisely the harmonic measure for \({{\mathbb {D}}}_e\) evaluated at the point \(\phi (z)\in {{\mathbb {D}}}_e\) (cf. [44]).
Pulling back the left and right hand sides in (4.37) by the inverse \(f_n^{1}(p,\ell )=(\phi (p),2\sqrt{n}\ell )\) and using conformal invariance of the harmonic measure, we infer that the measure \(\mu _{n,z}\) satisfies
and that the uniform convergence on the level of densities asserted in (1.29) holds. (The factor \(\tfrac{1}{\pi }\) in the left hand side comes from our normalization of the area measure dA.) q.e.d.
5 Proof of Lemma 4.1
In this section, we provide a detailed proof of Lemma 4.1 on tail kernel approximation, based on ideas from [54]. The main point is to give a derivation which leads to our desired estimates with minimal fuss, and in precisely the form that we want them. Aside from this, we believe that the following exposition could be of value for other investigations where the main interest is in leading order asymptotics.
When working out the details of this section, in addition to the original paper [54], we were inspired by [14], for example.
To briefly recall the setup, we take \(\{W_{j,n}\}_{j=0}^{n1}\) to be the orthonormal basis for the weighted polynomial subspace \({{\mathscr {W}}}_n\) of \(L^2\) with \(W_{j,n}=P_{j,n}\cdot e^{\frac{1}{2}nQ},\) where the polynomial \(P_{j,n}\) has degree j and positive leading coefficient. The tail kernel \({\tilde{K}}_n(z,w)\) is then given by
As always, we write
where U is the component of \(\hat{{{\mathbb {C}}}}\setminus S\) containing \(\infty \).
5.1 Reduction of the problem
Fix numbers \(\tau _0<1\), \(\epsilon >0\) and a compact subset \(K\subset {\text {Int}}\Gamma _{\tau _0\epsilon }\) with the properties in Sect. 3.2. Also fix j and n such that \(\tau _0\le \tau (j)\le 1\), where (as always) \(\tau (j)=j/n\).
Following [54] we define an approximation of \(W_{j,n}(z)\) on \({{\mathbb {C}}}\setminus K\) by
where
Here \({{\mathscr {H}}}_\tau \) and \({{\mathscr {Q}}}_\tau \) are bounded holomorphic functions on \(\hat{{{\mathbb {C}}}}\setminus K\) with \({\text {Re}}{{\mathscr {H}}}_\tau =\log \sqrt{\Delta Q}\) and \({\text {Re}}{{\mathscr {Q}}}_\tau =Q\) on \(\Gamma _\tau \); \(\phi _\tau \) is the univalent extension to \(\hat{{{\mathbb {C}}}}\setminus K\) of the normalized conformal map \(U_\tau \rightarrow {{\mathbb {D}}}_e\).
Lemma 5.1
(“Main approximation formula”). The number \(\tau _0<1\) may be chosen so that if \(\tau _0\le \tau (j)\le 1\) and if \(\beta \) is any number in the range \(0<\beta <\tfrac{1}{4}\) then as \(n\rightarrow \infty \),
It is clear that Lemma 5.1 implies Lemma 4.1 on asymptotics for the tail kernel \({\tilde{K}}_n(z,w)\).
The rest of this section is devoted to a proof of Lemma 5.1.
5.2 Foliation flow
One of the key ideas in [54] is to introduce a set of “flow coordinates” to facilitate computations.
In the following suppose that \(\tau _0\le \tau \le 1\); it will be convenient to write
Fix a small \(\delta >0\). For a small real parameter t, we denote by \(L_{\tau ,t}\) the level set
Of course \(L_{\tau ,0}=\Gamma _\tau \).
By Lemma 3.4 we see that for small \(t\ne 0\), \(L_{\tau ,t}\) is the disjoint union of two analytic Jordan curves \(L_{\tau ,t}=\Gamma _{\tau ,t}^\cup \Gamma _{\tau ,t}^+\) where \(\Gamma _{\tau ,t}^\subset {\text {Int}}\Gamma _\tau \) and \(\Gamma _{\tau ,t}^+\subset {\text {Ext}}\Gamma _\tau \). We set \(\Gamma _{\tau ,t}=\Gamma _{\tau ,t}^\) if \(t\le 0\) and \(\Gamma _{\tau ,t}=\Gamma _{\tau ,t}^+\) if \(t\ge 0\).
Let \(U_{\tau ,t}\) be the exterior domain of \(\Gamma _{\tau ,t}\) and consider the simply connected domain \(\phi _\tau (U_{\tau ,t})\subset \hat{{{\mathbb {C}}}}\).
Also denote by
the normalized conformal mapping (i.e., \(\psi _t(\infty )=\infty \) and \(\psi _t'(\infty )>0\)). Thus \(\psi _0(z)=z\) and \(\psi _t\) is to be regarded as a slight perturbation of the identity.
Note that \(\psi _t\) continues analytically across \({{\mathbb {T}}}\) and obeys the basic relation
Indeed, our definitions have been set up so that, for all large n,
With \(\tau (j)=j/n\), we define a neighbourhood \(D_{j,n}\) of \({{\mathbb {T}}}\) by
The inverse image \(\phi _{\tau (j)}^{1}(D_{j,n})\) plays the role of an “essential support” for \(W_{j,n}\) and \(W_{j,n}^\sharp \).
5.3 Approximation scheme
In the following we fix j and n with \(\tau _0\le \tau (j)\le 1\), where \(\tau (j)=j/n\). We shall extend \(W_{j,n}^\sharp \) to a smooth function on \({{\mathbb {C}}}\) by a straightforward cutoff procedure.
It is convenient to modify the compact set \(K\subset {\text {Int}}\Gamma _{\tau _0\epsilon }\) so that \(\phi _{\tau (j)}\) maps \(\hat{{{\mathbb {C}}}}\setminus K\) biholomorphically onto some exterior disc \({{\mathbb {D}}}_e(\rho _0\delta )\) where \(\rho _0<1\) and \(\delta >0\). (Then \(K=K(j)\) may slightly vary with j, but it will be harmless to suppress the jdependence in our notation.)
Next we fix a smooth function \(\chi _0\) such that \(\chi _0=0\) on K and \(\chi _0=1\) on \(\phi _{\tau (j)}^{1}({{\mathbb {D}}}_e(\rho _0))\) and define
(It is understood that \(W_{j,n}^\sharp =0\) on K.)
The following properties of the function \(W_{j,n}^\sharp \) are key for what follows:

(1)
\(W_{j,n}^\sharp \) is asymptotically normalized: \(\Vert W_{j,n}^\sharp \Vert =1+O({\varepsilon }_n)\) as \(n\rightarrow \infty \).

(2)
\(W_{j,n}^\sharp \) is approximately orthogonal to lower order terms: \((W,W_{j,n}^\sharp )\le Cn^{\frac{1}{2}}\Vert W\Vert \) for any \(W=P\cdot e^{\frac{1}{2}nQ}\in {{\mathscr {W}}}_n\) with \({\text {degree}}P<j\).
5.4 Positioning and the isometry property
Continuing in the spirit of [54], we define the “positioning operator” \(\Lambda _{j,n}\) by
Also define a function (“\(\tau (j)\)ridge”) by
where \(V_\tau \) is the harmonic continuation of \({\check{Q}}_\tau \Big _{U_\tau }\) inwards across \(\Gamma _\tau \).
The map \(\Lambda _{j,n}\) is then an isometric isomorphism
which preserves holomorphicity. (Here and in what follows, the norm in the weighted \(L^2\)space \(L^2_\phi (\Omega )\) is, by definition, \(\Vert f\Vert _\phi ^2=\int _\Omega f^2 e^{\phi }\, dA\).)
In particular we have the following “isometry property”,
We now define a function \(f_{j,n}\) on \({{\mathbb {D}}}_e(\rho _0)\) by
This gives
Lemma 5.2
With \(\delta _\tau (z)={\text {dist}}(z,\Gamma _\tau )\), we have for all \(z\in {{\mathbb {C}}}\setminus K\)
where C and c are positive constants.
Proof
Observe that
and use Lemma 3.5. \(\square \)
5.5 Integration in flowcoordinates
Define a domain \({\tilde{D}}_{n}\) in coordinates \((t,w)\in {{\mathbb {R}}}\times {{\mathbb {T}}}\) by
Now fix j with \(\tau _0\le \tau (j)\le 1\) and recall the definition of the flow domain \(D_{j,n}\) in (5.7).
Following [54] we define a flow map \(\Psi :{\tilde{D}}_n\rightarrow D_{j,n}\) by \(\Psi (t,w)=\psi _t(w)\), where \(\psi _t=\psi _{\tau (j),t}\). The Jacobian of the map \((t,w)\mapsto ({\text {Re}}\Psi ,{\text {Im}}\Psi )\) is calculated as
Lemma 5.3
As \(t\rightarrow 0\) we have
Moreover with \(f_{j,n}\) given by (5.10) we have
Proof
Write \(\psi _t(w)=w\cdot (1+t\hat{\psi }_1(w)+O(t^2))\). Since \(R_{\tau (j)}\circ \psi _t=t^2\) we obtain by Taylor’s formula (Lemma 3.4) that
so, since \({\text {Re}}\hat{\psi }_1(w)>0\),
and consequently
This proves (5.12); to prove (5.13) we set \(p=\phi _{\tau (j)}^{1}(w)\) and compute
\(\square \)
It follows that if f is an integrable function on \(D_{j,n}\) then
so by Lemma 5.3,
Taking \(f=f_{j,n}^2e^{nR_{\tau (j)}}\) and using that \(R_{\tau (j)}\circ \psi _t=t^2\) on \({{\mathbb {T}}}\), we now see that
Hence
and by Lemma 5.2, the last term on the right is \(O(\sqrt{n}e^{c\log ^2 n})\) for a suitable constant \(c>0\).
We have shown the approximate normalization property (1), i.e., we have shown:
Lemma 5.4
If \(\tau (j)\in [\tau _0,1]\) then \(\Vert W_{j,n}^\sharp \Vert =1+O({\varepsilon }_n)\) as \(n\rightarrow \infty .\)
5.6 Approximate orthogonality
We now prove property (2) of the quasipolynomials.
Given a positive integer k, it is convenient to write \({{\mathscr {W}}}_{k,n}\) for the space of weighted polynomials \(W=P\cdot e^{\frac{1}{2}nQ}\) where P has degree at most k, equipped with the usual \(L^2\)norm.
Lemma 5.5
Suppose that \(\tau _0\le \tau (j)\le 1\). Then for all \(W\in {{\mathscr {W}}}_{j1,n}\) we have
Proof
Let \(W=Pe^{\frac{1}{2}nQ}\) where P has degree \(\ell <j\). Write \(q=\Lambda _{j,n}^{1}[P]\); then q is holomorphic on \({{\mathbb {D}}}_e(\rho _0)\) and satisfies \(q(z)=O(z^{\ell j})\) as \(z\rightarrow \infty \).
By the CauchySchwarz inequality and Lemma 5.2 we conclude that
Hence it suffices to estimate the integral
where \(h=q/f_{j,n}\) is holomorphic of \({{\mathbb {D}}}_e(\rho _0)\) and vanishes at infinity (since \(f_{j,n}(\infty )>0\)).
By (5.15),
But
by the meanvalue property of holomorphic function, so we obtain the estimate
Since \(1/f_{j,n}\) is bounded on \({\tilde{D}}_{n}\) we see that
Using the CauchySchwarz inequality the right hand side is estimated by
where we used the isometry property (5.9) to deduce the equality. \(\square \)
5.7 Pointwise estimates
We wish to show that when \(\tau (j)\) is close to 1, then \(W_{j,n}\) is “pointwise close” to \(W_{j,n}^\sharp \) near the curve \(\Gamma _{\tau (j)}\).
Lemma 5.6
There are constants C and \(n_0\) such that for all \(n\ge n_0\) and all j with \(\tau _0\le \tau (j)\le 1\) we have \(\Vert W_{j,n}W_{j,n}^\sharp \Vert \le C{\varepsilon }_n.\)
Proof
Let \(u_0\) be the normminimal solution in \(L^2(e^{nQ},dA)\) to the following \(\bar{\partial }\)problem:

(i)
\(\bar{\partial }u=F_{j,n}\cdot \bar{\partial }\chi _0\) on \({{\mathbb {C}}}\),

(ii)
\(u(z)=O(z^{j1})\) as \(z\rightarrow \infty \).
A standard estimate found in [57, Section 4.2] shows that there is a constant C such that
Since \(\bar{\partial }\chi _0=0\) on \(U_{\tau _0\epsilon }\), Lemma 5.2 implies that there is a constant \(c>0\) such that \(F_{j,n}^2e^{nQ}\le e^{cn}\) on the support of \(\bar{\partial }\chi _0\). Thus
with (new) positive constants C and c.
We correct \(F_{j,n}\cdot \chi _0\) to a polynomial \({\tilde{P}}_{j,n}\) of exact degree j by setting
(\({\tilde{P}}_{j,n}\) is then an entire function of exact order of growth \(O(z^j)\) as \(z\rightarrow \infty \), since \(F_{j,n}(z)\asymp z^j\) and \(u_0(z)\lesssim z^{j1}\) as \(z\rightarrow \infty \), so indeed \({\tilde{P}}_{j,n}\) is a polynomial of exact degree j.)
It follows from (5.17) that
Recall that \(W_{j,n}^\sharp =\chi _0\cdot F_{j,n}\cdot e^{\frac{1}{2}nQ}\), set \({\tilde{W}}_{j,n}={\tilde{P}}_{j,n}\cdot e^{\frac{1}{2}nQ}\), and note that (5.18) says that
By Lemma 5.4 we have \(\Vert W_{j,n}^\sharp \Vert =1+O({\varepsilon }_n)\) and so by (5.19),
Similarly, the approximate orthogonality in Lemma 5.5 implies (with the estimate (5.18)) that
Now let \(\pi _{j1,n}:L^2\rightarrow {{\mathscr {W}}}_{j1,n}\) be the orthogonal projection and put \(W_{j,n}^*={\tilde{W}}_{j,n}\pi _{j1,n}({\tilde{W}}_{j,n}).\) Then \(\Vert {\tilde{W}}_{j,n}W_{j,n}^*\Vert =\Vert \pi _{j1,n}({\tilde{W}}_{j,n})\Vert =O(n^{\frac{1}{2}})\) by (5.21), and so
Moreover, since \(W_{j,n}^*\in {{\mathscr {W}}}_{j,n}\ominus {{\mathscr {W}}}_{j1,n}={\text {span}}\{W_{j,n}\}\), we can write \(W_{j,n}^*=c_{j,n}W_{j,n}\) for some constant \(c_{j,n}\), which we can assume is positive. Since \(\Vert W_{j,n}\Vert =1\) we then have \(c_{j,n}=1+O({\varepsilon }_n)\). It follows that
and the proof of the lemma is complete. \(\square \)
Following a wellknown circle of ideas we shall now turn the \(L^2\)estimate in Lemma 5.6 into a pointwise one.
Lemma 5.7
Suppose \(\tau _0\le \tau (j)\le 1\) and that u is a smooth function on \({{\mathbb {C}}}\) which is holomorphic in \({{\mathbb {C}}}\setminus K\) with \(u(z)\lesssim z^{j}\) as \(z\rightarrow \infty \). Consider the weighted analytic function \(W=u\cdot e^{\frac{1}{2}nQ}\) on \({{\mathbb {C}}}\setminus K\). Then there exists a constant C such that
Proof
We shall slightly modify our proof of Lemma 3.7. Write \(\tau =\tau (j)\). We begin by recording the basic estimate
In order to verify (5.22), we may assume that \(M_\tau \le 1\). The estimate is trivial if \(z\not \in U_\tau \) so we may also assume that \(z\in U_\tau \).
We then form the function
By assumption, this function is subharmonic on \({{\mathbb {C}}}\setminus K\) and satisfies \(s\le Q\) on \(\Gamma _\tau \). Moreover, we know that \(s(z)\le 2\tau \log z+O(1)\) as \(z\rightarrow \infty \). Hence \(s\le {\check{Q}}_\tau \) on \({{\mathbb {C}}}\setminus U_\tau \) by the strong version of the maximum principle.
Next pick \(n_0\) such that Q is smooth in a neighbourhood of \(({\text {cl}}U_{\tau _0})\setminus U_\tau \). Also fix an arbitrary point \(w\in ({\text {cl}}U_{\tau _0})\setminus U_\tau \).
By [5, Lemma 2.4 and its proof], we have
where C is independent of w. (Indeed, C depends only on the maximum of the Laplacian \(\Delta Q\) over a slightly enlarged set.)
The lemma is immediate on combining (5.22) and (5.23). \(\square \)
5.8 Proof of Lemma 5.1
Fix a number \(\beta \in (0,\tfrac{1}{4})\) and suppose that \(z\in N(U_{\tau (j)},\delta _n)\), where \(\tau (j)\) is in the interval \([\tau _0,1]\).
By (a straightforward generalization of) the inequality (4.30) we have the estimate
Applying Lemma 5.7 and Lemma 5.6 with \(W=W_{j,n}W_{j,n}^\sharp \) we obtain
But by definition of \(F_{j,n}\) it is clear that
Since \(\beta <\frac{1}{4}\), \(n^{\frac{1}{4}}\) outgrows \(n^{\beta }(\log n)^{CM^2+1}\) as \(n\rightarrow \infty \), so it follows from (5.25) and (5.26) that
Our proof of Lemma 5.1 is complete. \(\square \)
6 The Loop Equation and Complete Integrability
In this section we view the Berezin measures as exact solutions to the loop equation and we briefly discuss the imposed integrable structure on the coefficients in the corresponding large n expansion of the onepoint function.
An advantage of the loop equation point of view is that it continues to hold in a context of \(\beta \)ensembles, thereby making it potentially useful for the study of the Hall effect, freezing problems and related issues of interest in contemporary mathematical physics.
We shall not attempt a profound analysis here; we will merely point out how the loop equation fits in with some of our work in the previous sections. More about the use of loop equations and large nexpansions can be found in the papers [5, 11, 12, 16, 22, 30, 31, 39, 53, 59, 78] and the references there.
6.1 Gaussian approximation of harmonic measure as a solution to the loop equation
Let \(K_n(z,w)\) be the reproducing kernel with respect to an admissible potential Q. We will write \(R_n(z)=K_n(z,z)\) for the 1point function and \(R_{n,k}(w_1,\ldots ,w_k)=\det (K_n(w_i,w_j))_{k\times k}\) for the kpoint function of the determinantal Coulomb gas process \(\{z_j\}_1^n\) associated with Q.
By Theorem 1.5 we know that if z is in the exterior domain U, then the Berezin measure \(\mu _{n,z}\) obeys the asymptotic
where \(\omega _z\) and \(\gamma _{n,p}\) denote certain harmonic and Gaussian measures, respectively.
As we shall see (whether or not z is in the exterior) \(\mu _{n,z}\) is an exact solution to the loop equation
where \(k_z(w)\) is the Cauchy kernel
(We remind that \(\mu (f)\) is short for \(\int f\, d\mu \).)
The relation (6.2) is not the “usual” form of the twodimensional loop equation (e.g. [11]), but rather a kind of infinitesimal variant; for completeness we include a derivation of it below.
6.2 \(\beta \)ensembles
Given a large n and a configuration \(\{z_j\}_1^n\) we consider the Hamiltonian
The BoltzmannGibbs law in external potential Q and inverse temperature \(\beta \) is the following probability law on \({{\mathbb {C}}}^n\),
Suppose that \(\{z_j\}_1^n\) is picked randomly with respect to (6.3). For fixed \(k\le n\) we denote by \(R_{n,k}^\beta \) the kpoint function, i.e., the unique (continuous) function on \({{\mathbb {C}}}^k\) obeying
for each bounded Borel function f on \({{\mathbb {C}}}^k\).
We shall also use the connected 2point function \(R_{n,2}^{\beta ,(c)}\), which is defined by
Finally, we introduce the Berezin kernel \(B_n^\beta (z,w)\) and the Berezin measure \(\mu _{n,z}^\beta \) by
In the following we denote by \(R_n^\beta =R_{n,1}^\beta \) the 1point function.
6.3 Proof of the loop equation
We have the following variant of the loop equation. (See e.g. [12, 22, 31, 59, 78] and references for related identities).
Proposition 6.1
If Q is \(C^2\)smooth in a neighbourhood of a point z, then
Proof
Let \(\Lambda \subset {{\mathbb {C}}}\) be an open set such that Q is \(C^2\)smooth in a neighbourhood of the closure \({\text {cl}}\Lambda \). Fix a point \(z\in \Lambda \) and a smooth realvalued function \(\psi \) supported in \(\Lambda \).
Given a random sample \(\{z_j\}_1^n\), we can for each j view the number \(\psi (z_j)\) as a random variable with respect to (6.3).
We use integration by parts to see that, for each fixed j
(In the last expression, \({\partial }_j\) is short for \({\partial }/{\partial }z_j\).)
Now observe that for each j
Hence a summation in j gives
We have shown that
where \(W_n^+[\psi ]\) is the “Ward’s tensor”
The identity (6.4) is what is called “Ward’s identity” in papers such as [12]. To deduce the infinitesimal version in Proposition 6.1 we proceed as follows.
By the definition of 1point function and an integration by parts we have
Also
and
Since the identity (6.4) holds for every testfunction \(\psi \), we obtain the pointwise identity for all \(z\in \Lambda \),
(First we obtain the identity in the sense of distributions on \(\Lambda \), then everywhere, since the functions involved are smooth.)
Dividing through by \(\beta R_{n,1}^\beta \) we find
Recalling that
we obtain
Taking \(\bar{\partial }\)derivatives with respect to z in the last identity, we finish the proof. \(\square \)
6.4 On various asymptotic relations
Now set \(\beta =1\). We have the following theorem on the Cauchy transform \(\mu _{n,z}(k_z)\) of the Berezin measure \(\mu _{n,z}\). (The symbol \(k_z\) denotes the Cauchy kernel \(k_z(w)=(zw)^{1}\), and \(\omega _z\) denotes the harmonic measure of U evaluated at a point \(z\in U\).)
Theorem 6.2
With \(\phi :U\rightarrow {{\mathbb {D}}}_e\) the normalized conformal map, we have the identity
where H(z) is a holomorphic function in U with \(H(z)=O(z^{2})\) as \(z\rightarrow \infty \).
Proof
The first equality in (6.6) is immediate by Theorem 1.5. (The singularity of \(k_z(w)\) at \(w=z\) presents no trouble since \(B_n(z,w)\le K_n(w,w)\) and \(K_n(w,w)\) converges to zero uniformly for w outside of any given neighbourhood of \(({\text {Int}}\Gamma )\cup \Gamma \), see for example [5, Theorem 1].)
In order to prove the remaining equality, we assume for simplicity that Q is \(C^2\)smooth throughout \(U\setminus \{\infty \}\); the extension to more general admissible potentials may be left to the reader.
We shall use Theorem 1.3, which implies that for \(z\in U\),
where \(0<\beta <\frac{1}{4}\). This implies that \(R_n(z)\) is negligible for large n whereas
It follows that
and hence by Proposition 6.1, we have with locally uniform convergence
Hence the function \(H(z)=\lim \limits _{n\rightarrow \infty }\mu _{n,z}(k_z)\frac{{\partial }}{{\partial }z}\log (\phi (z)^21)\) is holomorphic in U, and since (for each n)
we see that \(H(z)=O(z^{2})\) as \(z\rightarrow \infty \). \(\square \)
Example
Suppose that Q(z) is radially symmetric. Then \(U={{\mathbb {D}}}_e(r)\) is an exterior disc, and the conformal map \(\phi :U\rightarrow {{\mathbb {D}}}_e\) is just \(\phi (z)=z/r\); the harmonic measure is \(dP_z(\theta )\, d\theta \) where \(P_z(\theta )=\frac{1}{2\pi }\frac{z^2r^2}{zre^{i\theta }^2}\) for \(z>r\). A standard computation gives that \(\omega _z(k_z)=\frac{{\bar{z}}}{z^2r^2}=\frac{{\partial }}{{\partial }z}\log (\phi (z)^21)\). Hence H in (6.6) vanishes identically if Q is radially symmetric. On the other hand, for nonsymmetric potentials, H(z) seems typically to be nontrivial.
For the Ginibre ensemble we have the following result.
Theorem 6.3
When \(Q(z)=z^2\), we have the asymptotic expansion
Before proving the theorem, we remark that we can at this point easily prove a differentiated form of (6.8). Namely, by Theorem 1.1, we know that
where \(\rho _1(\zeta )=\frac{1}{12}\frac{\zeta }{(\zeta 1)^2}\). Passing to logarithms and differentiating we see that, for \(z>1\),
Since \(R_n(z)\) is negligible for \(z>1\) while \(\Delta Q=1\), we find (after some computation) that the right hand side in Proposition 6.1 equals to
One checks readily that the right hand side in (6.9) equals to the \({\partial }/{\partial }{\bar{z}}\)derivative of (6.8).
Proof of Theorem 6.3 (Sketch)
For fixed \(z\in {{\mathbb {D}}}_e\) we write \(k_z(w)=(zw)^{1}\) for the Cauchykernel and \(d\omega _z(\theta )=P_z(\theta )\, d\theta \) for the harmonic measure of \(U={{\mathbb {D}}}_e\) evaluated at z. Here of course \(P_z(\theta )=\frac{1}{2\pi }\frac{z^21}{ze^{i\theta }^2}\) is the exterior Poisson kernel.
By Theorem 6.2 we already know that
To find the O(1/n)term in (6.8), we fix \(z\in {{\mathbb {D}}}_e\) and use the approximation provided by Theorem 1.1,
and work in the coordinate system \((\theta ,t)\) where \(w=e^{i\theta }\cdot (1+\frac{t}{2\sqrt{n}})\). A lengthy but straightforward computation based on Taylor’s formula, residues, and the elementary identity \(\omega _w(k_z)=\frac{{\bar{w}}}{z{\bar{w}}1}\) produces the asymptotic in (6.8); we omit details. \(\square \)
Beyond the Ginibre ensemble, it is not clear from our above results that there is a similar large nexpansion. However, a qualitative result of Hedenmalm and Wennman comes to the rescue.
Theorem 6.4
Let Q be a potential satisfying the assumptions in Subsection 1.2. There is then an asymptotic expansion
where \(z\in U\) and \(\rho _1,\rho _2,\ldots \) are some unknown correction terms which are subject to the completely integrable system given by Proposition 6.1.
Proof
Fix z in the exterior component U and pick w near \(\Gamma \) or in U. It follows from [53, Theorem 1.4.1] that there is an asymptotic expansion
where \(F_n(w,z)\) and the leading term \(a_0(w,z)\) are explicitly given in [53].
It follows from (6.11) that
where \(F_n(w,z)^2\) and \(b_0(z,w)=a_0(w,z)^2\) are again certain explicit functions, which of course must match up with the expressions found in Theorem 1.5 for \(z\in U\) and w near the boundary \({\partial }U\).
In the expansion (6.11), the points z and w play highly asymmetric roles. Nevertheless, the form of the expansion (6.12), specialized to the diagonal case when \(w=z\) belongs to U shows that the form of (6.10) must hold for appropriate correction terms \(\rho _j(z,{\bar{z}})\), which are proportional to \(b_j(z,z)/b_0(z,z)\). \(\square \)
By inserting the ansatz (6.10) in the loop equation, we get a feedback relation for the correction terms \(\rho _j(z,{\bar{z}})\). A deeper analysis of this structure is beyond the scope of our present investigation.
6.5 Back to \(\beta \)ensembles
We finish with a few words about \(\beta \)ensembles. In the case when \(z\in U\), it is known due to the localization theorem in [5] that \(R_n^\beta (z)\rightarrow 0\) quickly as \(n\rightarrow \infty \). Hence the right hand side in Ward’s identity (Proposition 6.1) is
(We assume here that Q is smooth at z.)
The left hand side in Ward’s identity is not known, but it seems plausible that we should have \(\frac{{\partial }}{{\partial }{\bar{z}}}(\mu _{n,z}^\beta (k_z))=O(1)\) when \(z\in U\). Assuming that this is the case, and comparing O(n)terms in Ward’s identity we find “heuristically” the approximation
where the dots represent terms of lower order in n.
When \(z\in U\) is close to the boundary \({\partial }U\), (6.13) is consistent with predictions found in [30], and also with the localization theorem in [5].
The papers [16, 30, 31] and the references there provide more information about the problem of finding asymptotics for the 1point function \(R_n^\beta \) when \(\beta >1\).
6.6 A glance at disconnected droplets
We now briefly touch on the case of disconnected droplets, where the condition (1) in the definition of an admissible potential is replaced by

(1’)
Q is \(C^2\)smooth on S and real analytic in a neighbourhood of the outer boundary \(\Gamma ={\partial }U\).
We start by noting that if we replace assumption (1) by (1’) in our definition of admissible potential, then the existence of a local Schwarz function \({{\mathscr {S}}}\) at each point \(p\in \Gamma \) can be established precisely as in the proof of Lemma 3.1. It follows by Sakai’s main result in [70] that \({{\mathbb {C}}}\setminus U\) has finitely many components \(K_1,\ldots ,K_d\), and that the normalized Riemann maps \(\chi _l:{{\mathbb {D}}}_e\rightarrow \hat{{{\mathbb {C}}}}\setminus K_l\) can be continued analytically across \({{\mathbb {T}}}\) for \(l=1,\ldots ,d\). We put \(\Gamma ^l=\chi _l({{\mathbb {T}}})={\partial }K_l\) and assume that \(\chi _l'\ne 0\) on \({{\mathbb {T}}}\) for \(l=1,\ldots ,n\). Then each \(\Gamma ^l\) is an analytic, nonsingular Jordan curve.
For a basic model case we consider the disconnected lemniscate droplet, defined by the potential
where \(d\ge 2\) is an integer.
By [26, Lemma 1], the \(\tau \)droplet (i.e. droplet in potential \(Q/\tau \)) is
See Fig. 8.
We write \(S=S_1\) and note that the the equilibrium measure is \(d\sigma (z)=dz^{2(d1)}{\textbf{1}}_S(z)\, dA(z).\)
The following problem presents itself: to find the asymptotics of \(K_n(z,w)\) when z and w belong to different boundary components of \({\partial }S\). Since U is not simply connected, there is no longer a Riemann map \(\phi \), and as a consequence our technique using quasipolynomial approximation will not work, at least not without substantial changes. Fortunately, in the special case of the potential (6.14), approximate orthogonal polynomials can be found using RiemannHilbert techniques following the works [18, 19, 29, 63]. This was used to generate Fig. 9.
The recent work [32] studies other types of ensembles with disconnected droplets, with “hard edges”, in which the droplet consists of several concentric annuli. Among other things it is shown that a Jacobi theta function emerges when studying certain associated gapprobabilities. (More generally, theta functions are known to emerge in various more or less related contexts, see [25] and the references there.) The present setting of “soft edge” ensembles with disconnected droplets is the topic of our forthcoming work [8].
Other types of open problems enter in the case when the boundary of the droplet has one or several singular points (cusps, double points, or lemniscatetype singularities which are found at boundary points where \(\Delta Q=0\)). More background on singular points can be found e.g. in [13] and the references there.
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Ameur, Y., Cronvall, J. Szegő Type Asymptotics for the Reproducing Kernel in Spaces of FullPlane Weighted Polynomials. Commun. Math. Phys. 398, 1291–1348 (2023). https://doi.org/10.1007/s0022002204539y
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DOI: https://doi.org/10.1007/s0022002204539y