Szeg\H{o} type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials

In this work we find and discuss an asymptotic formula, as $n\to\infty$, for the reproducing kernel $K_n(z,w)$ in spaces of full-plane weighted polynomials $W(z)=P(z)\cdot e^{-\frac 12nQ(z)},$ where $P(z)$ is a holomorphic polynomial of degree at most $n-1$ and $Q(z)$ is a fixed, real-valued function termed"external potential". The kernel $K_n$ corresponds precisely to the canonical correlation kernel in the theory of random normal matrices. As is well-known, the large $n$ behaviour of $K_n(z,w)$ must depend 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 $Q$. In the particular case when $z$ and $w$ are at the edge and $z\ne w$, we prove the formula $K_n(z,w)\sim\sqrt{2\pi n}\,\Delta Q(z)^{\frac 1 4}\Delta Q(w)^{\frac 14}\,S(z,w)$ where $S(z,w)$ is the Szeg\H{o} kernel associated with the Hardy space $H^2_0(U)$ of analytic functions on unbounded component $U$ of $\hat{\mathbb{C}}\setminus S$ which vanish at infinity. 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.

To arrive at this kernel, we are prompted to equip C with the background measure The law of {z j } n 1 is the Gibbs measure where dA n = (dA) ⊗n is the normalized Lebesgue measure on C n .(The combinatorial factor 1/n! accounts for the fact that elements (z j ) n 1 ∈ C n are ordered sequences, while configurations {z j } n The expected number of particles which fall in a given Borel set E is and if f (z 1 , . . ., z k ) is a compactly supported Borel function on C k where k ≤ n, then where the k-point function R n,k (w 1 , . . ., w k ) = det(K n (w i , w j )) k i,j=1 .We reserve the notation for the 1-point function.
We shall here study the case when |z w − 1| ≥ η for some η > 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 so-called Szegő kernel emerges in the off-diagonal 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 sz to be the unbounded component of C \ γ sz , i.e., E sz = Ext γ sz .
(See Figure 1.) Figure 1.The exterior Szegő domain E sz in grey.
1.1.1.Szegő type asymptotics for the Ginibre kernel.Three principal cases emerge, depending on the location of the product z w.
(i) If z w ∈ C \ (E sz ∪ {1}) we have bulk type asymptotic in the sense that (Cf.Subsection 2.2 for more about this.) (ii) If z w is in a microscopic neighbourhood of z w = 1, then (1.1) has a well-understood error-function asymptotic given in [12,Subsection 2.2].Further results in this direction can be found in [26,54,73], for example.(iii) If z w ∈ E 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 w ∈ E sz and let K n (z, w) be the Ginibre kernel (1.1).Then as n → ∞ (1.4) The O-constant is uniform provided that ζ = z w remains in a compact subset of E sz ; the correction term ρ j (ζ) is a rational function having a pole of order 2j at ζ = 1 and no other poles in the extended complex plane Ĉ = C ∪ {∞}; the first one is given by and the higher ρ j (ζ) can be computed by a recursive procedure based on (1.34), (1.35) below.
In the case when ζ = z w belongs to the sector | arg(ζ − 1)| < 3π 4 , the result can alternatively be deduced by writing the kernel as a product involving an incomplete gamma-function 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 sz where the same asymptotic formula applies.See Subsection 1.5 for further details.
For k = 0, Theorem 1.1 implies that (1.5) • (1 + o(1)), (z w ∈ E sz ).Now assume that both z and w are on the unit circle T = ∂S = {|z| = 1}.In this case the function c n (z, w) = z n wn = 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 "∼" to mean "up to cocycles", (1.5) implies Let D e = {|z| > 1} ∪ {∞} be the exterior disc and dθ = |dz| the arclength measure on T. Consider the Hardy space H 2 0 (D e ) of analytic functions f : D e → C which vanish at infinity, equipped with the norm of L 2 (T, dθ).The kernel S(z, w) is the reproducing kernel of H 2 0 (D e ).Let us now consider the Berezin kernel rooted at a point z ∈ C, (1.8) B n (z, w) = |K n (z, w)| 2 K n (z, z) .It is interesting to compare (1.9) with the case when z ∈ Int S; then B n (z, w) decays exponentially in n by the heat-kernel estimate in Subsection 2.2.(Alternatively, by results in [9].) The moral is that, in the off-diagonal case z ̸ = w, the magnitude of K n (z, w) is exceptionally large when both z and w are on the boundary T, compared with any other kind of configuration.(Some heuristic explanations for this kind of behaviour are sketched below in Subsection 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 µ n,z rooted at z, (1.10) dµ n,z (w) = B n (z, w) dA(w).
It is shown in [9,Section 9] that if z ∈ D e then the measures µ n,z converge weakly to the harmonic measure relative to D e evaluated at z, dω z (θ) = P z (θ) dθ where P z (θ) is the (exterior) Poisson kernel (1.11) P z (θ) = 1 2π We will denote by dγ n the following Gaussian probability measure on R (Here and throughout, "dℓ" is Lebesgue measure on R.) It is also convenient to represent points w close to T in "polar coordinates" (1.12) w = e iθ • (1 + ℓ), (θ ∈ [0, 2π), ℓ ∈ R).
As a consequence of the kernel asymptotic in Theorem 1.1, we obtain the following result.
Corollary 1.2.Fix a point z ∈ D e and an arbitrary sequence (c n ) ∞ 1 of positive numbers with nc 2 n → ∞, and nc 3 n → 0, as n → ∞.
Then for w in the form (1.12), we have the Gaussian approximation Here and henceforth, a sequence of functions f n : E n → C is said to converge uniformly to 0 if there is a sequence ϵ n → 0 such that |f n | < ϵ n on E n for each n.
Remark.The approximating measures dμ n,z (θ, ℓ) = P z (θ) • γ n (ℓ) dθdℓ are probability measures on T × R which assign a mass of O(e −nc 2 n ) to the complement of N (T, c n ).The condition that nc 2  n → ∞ insures that µ n,z − μn,z → 0 in the sense of measures on C.This justifies the Gaussian approximation picture, as exemplified in Figure 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 : C → R ∪ {+∞} which we call the external potential.(The Ginibre ensemble corresponds to the special choice 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 µ, we define its Q-energy by (1.15) where µ(Q) is short for Q dµ.
It is well-known [69] that there exists a unique equilibrium measure σ = σ Q of unit mass which minimizes I Q [µ] over all compactly supported Borel probability measures on C. The support of σ is denoted by S = S[Q] = supp σ, and is called the droplet.Perhaps even more central to this work is the exterior component containing ∞, The boundary of U is called the outer boundary of S and is written Γ = ∂U.
(1) S is connected and Q is C 2 smooth in a neighbourhood of S and real-analytic in a neighbourhood of Γ.
This assumption has the consequence that the equilibrium measure σ is absolutely continuous and has the structure dσ = 1 S •∆Q dA, where ∆ is the normalized Laplacian.We are guaranteed that ∆Q ≥ 0 on S; we will require a bit more: (2) ∆Q(z) > 0 for all z ∈ Γ.
Let SH 1 (Q) denote the class of all subharmonic functions s(z) on C which satisfy s ≤ Q on C and s(z) ≤ log |z| 2 + O(1) as z → ∞.We define the obstacle function Q(z) to be the envelope Clearly Q(z) is subharmonic and grows as log |z| 2 + O(1) as z → ∞.Furthermore, Q(z) is C 1,1 -smooth on C, i.e., its gradient is Lipschitz continuous.
Denote by S * = {z ; Q(z) = Q(z)} the coincidence set for the obstacle problem.In general we have the inclusion S ⊂ S * and if p is a point of S * \ S then there is a neighbourhood N of p such that σ(N ) = 0. We impose: (3) U ∩ S * is empty.
Write χ : D e → U for the unique conformal mapping normalized by the conditions χ(∞) = ∞ and χ ′ (∞) > 0. A fundamental theorem due to Sakai [70] implies that χ extends analytically across Γ to some neighbourhood of the closure cl D e .(Details about this application of Sakai's theory are found in Subsection 3.1 below.)Thus Γ is a Jordan curve consisting of analytic arcs and possibly finitely many singular points where the arcs meet.We shall assume: (4) Γ is non-singular, i.e., χ extends across T to a conformal mapping from a neighbourhood of cl D e to a neighbourhood of cl U .
In the following we denote by ϕ = χ −1 the inverse map, taking a neighbourhood of cl U conformally onto a neighbourhood of cl D e , and obeying ϕ(∞) = ∞ and ϕ ′ (∞) > 0. We denote by √ ϕ ′ the branch of the square-root which is positive at infinity.
Class of admissible potentials.Except when otherwise is explicitly stated, all external potentials Q used below are lower semicontinuous functions C → R ∪ {+∞}, finite on some set of positive capacity, satisfying the growth condition (1.14) and the four conditions ( 1)-( 4).
Auxiliary functions.For a given admissible potential Q, we consider the holomorphic functions Q(z) and H (z) on a neighbourhood of cl U which obey and which satisfy Im We shall also frequently use the function V given by (1.18) V = "the harmonic continuation of the restriction Q U across the analytic curve Γ." It is useful to note the identity where K is a fixed compact subset K of the bounded component Int Γ of C \ Γ.
To realize (1.19) it suffices to note that the harmonic functions on the left and right hand sides agree on Γ 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]).
The Szegő kernel.Let H 2 0 (U ) be the Hardy space of holomorphic functions f : U → C which vanish at infinity and are square-integrable with respect to arclength: Γ |f (z)| 2 |dz| < ∞.We equip H 2 0 (U ) with the inner product of L 2 (Γ, |dz|) and observe that the functions We shall refer to S(z, w) as the Szegő kernel associated with Γ (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].
The reproducing kernel.Let Q be an admissible potential and consider the space W n = W n (Q) consisting of all weighted polynomials W of the form where P is a holomorphic polynomial of degree at most n − 1.We equip W n with the usual norm in L 2 (C, dA) and denote by K n (z, w) the corresponding reproducing kernel.We follow standard conventions concerning reproducing kernels [16]; we write K n,z (w) = K n (w, z) and note that the element K n,z ∈ W n is characterized by the reproducing property: We shall frequently use the formula where {W j,n } n−1 j=0 is any orthonormal basis for W n .We fix such a basis uniquely by requiring that W j,n = P j,n • e − 1 2 nQ where P j,n is of exact degree j and has positive leading coefficient.
Auxiliary regions.In the sequel we write where M is a fixed positive constant (depending only on Q).The δ n -neighbourhood of a set E will be denoted where D(a, r) = {z ; |z − a| < r} is the euclidean disc with center a and radius r.
we have the asymptotic formula (1.23) The O-constant is uniform for the given set of z and w (depending only on the parameters η, M and the potential Q).
In the off-diagonal case when z, w are exactly on the boundary, we recognize several exact cocycles which may be cancelled 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.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 non-zero 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 ̸ = 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 Subsection 1.4 below.
Remark.It is interesting to view the slow decay of charge-charge 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 2-point 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 β-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 β = 2 and was settled in [11,68].)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 non-degenerate bulk point (in the sense that z ∈ Int S and ∆Q(z) > 0), then µ n,z converges to the Dirac point mass δ z , whereas if z ∈ U , then µ n,z converges to the harmonic measure ω z evaluated at z; the convergence holds in the weak sense of measures on C. (See [10, Theorem 7.7.2].) (2) If z is a non-degenerate bulk-point, then the convergence µ n,z → δ z is Gaussian in the sense of heat-kernel asymptotic: ), where o(1) → 0 uniformly for (say) w ∈ D(z, δ n ).(See e.g.[9].) (3) If z ∈ U , then the weak convergence µ n,z → ω z may be combined with an asymptotic result for the so-called root-function 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 µ n,z from which the convergence to harmonic measure will be directly manifest.For this purpose we express points w in some neighbourhood of Γ as where p = p(w) is a point on Γ, n 1 (p) is the unit normal to Γ pointing outwards from S, and ℓ is a real parameter.(So |ℓ| = dist(w, Γ) if ℓ is close to 0.) Given a point p ∈ Γ we also define a Gaussian probability measure γ p,n on the real line by For a given point z ∈ U , we denote by ω z the harmonic measure of U evaluated at z and consider the measure μn,z given in the coordinate system (1.25) by (1.27) dμ n,z = dω z (p) dγ n,p (ℓ).
To be more explicit, we define the Poisson kernel P z (p) as the density of ω z with respect to arclength |dp| on Γ, i.e., Then For this definition to be consistent, we fix a small neighbourhood of Γ and define μn,z by (1.28) in this neighbourhood and extend it by zero outside the neighbourhood.Then μn,z is a sub-probability measure whose total mass quickly increases to 1 as n → ∞.
Theorem 1.5.Suppose that z is in the exterior component U .Then where o(1) → 0 as n → ∞ with uniform convergence when w is in the belt N (Γ, In other words, µ n,z = (1 + o(1))μ n,z where o(1) → 0 as n → ∞ in the sense of measures on C as well as in the uniform sense of densities on N (Γ, δ n ).
Remark.The last statement in Theorem 1.5 is automatic once the uniform convergence in (1.29) is shown.Indeed, let ϵ > 0 be given.It is clear from the definition (1.28) that we can find M and n 0 such that μn,z (N (Γ, δ n )) > 1 − ϵ when n ≥ n 0 .Then μn,z (C \ N (Γ, δ n )) < ϵ when n ≥ n 0 .Thus μn,z | C\N (Γ,δn) → 0 as measures when n → ∞.By the uniform convergence in (1.29) we now see that µ n,z | C\N (Γ,δn) → 0, since µ n,z has unit total mass.Thus it suffices to prove the uniform convergence in (1.29); this is done in Section 4. 1.3.3.Main strategy: tail-kernel 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 leading-order asymptotics of the full kernel K n (z, w).Note that Kn (z, w) is just the reproducing kernel for the orthogonal complement Wn = W n ⊖ W k,n where W k,n ⊂ W n is the subspace consisting of all W = P • e − 1 2 nQ where P has degree at most k = "largest integer which is strictly less than nθ n ".
We shall deduce asymptotics for Kn (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 ≤ nθ n ) are sufficient to show that the full kernel K n (z, w) has similar asymptotic properties as does Kn (z, w).
The practical execution of this strategy forms the bulk of this paper, cf.sections 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 "non-Hermiticity parameter" τ with −1 < τ < 1 and set the parameters in (1.31) to a = 1 1+τ and b = 1 1−τ , giving ).However, other conventions are sometimes used, e.g.[62] takes a = 1 − τ and b = 1 + τ while [7] fixes a = 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 × n GUE matrices J 1 and J 2 and look at the random matrix The eigenvalues {z j } n 1 of X n then correspond precisely to a random sample from the determinantal n-point process in potential Q; this is what was used to produce Figure 3.
We now recast some well-known facts about the elliptic Ginibre point-process; 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) j e z 2 d j dz j e −z 2 , the correlation kernel K n (z, w) is given by (This formula was used to plot Figure 4.)Moreover, the droplet is the elliptic disc S = {z = x + iy ; a 2 +ab 2b x 2 + ab+b 2 2a y 2 ≤ 1}, which has its major semi-axis along the real line.The normalized conformal map ϕ taking U = Ĉ \ S to D e is the inverse Joukowsky map (well-known from the theory of conformal mapping [65]) ). (Here we use the principal branch of the square-root, so ϕ(z) ∼ z/α as z → ∞ and ϕ ′ (∞) = 1/α.) Since the Laplacian ∆Q is the constant 1 2 (a + b), we have H ≡ 1 2 log(a + b).Inserting these data, our Theorem 1.3 (and using Re Q = Q on ∂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 off-diagonal 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, n-independent 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, δ n ), of distance δ 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 line-segment between the foci of the ellipse, and such that |ϕ(z)ϕ(w) − 1| ≥ η for some η > 0, again see [4,Remark I.4].In particular, this provides information about the transition from exterior to bulk-type 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 = +∞ on C \ R we obtain Hermitian random matrix theory and Coulomb gas processes on 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 [6,12,54,53,62] and the references there.
It is noteworthy that Forrester and Honner in the paper [41] study a different problem on edge-correlations, between zeros of random polynomials p n (z) = n−1 j=0 (j!) − 1 2 a j z j where the a j are i.i.d.standard complex Gaussians.(The "edge" here is the circle |z| = √ n.) Szegő's paper [72] concerns zeros of partial sums S n (z It is not surprising that Szegő's results should have a bearing for the Ginibre ensemble, since a factor S n−1 (nz 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 Subsection 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].
Using the form in [67] we obtain readily that if where b 0 (ζ) = 1 and Corollary 1.6.The asymptotic expansion holds for all ζ in the exterior Szegő domain E sz .The domain E sz is moreover the largest possible domain in which the expansion (1.36) holds.
Remark.The complete large n asymptotics of Γ(n, nζ) for ζ in the complex plane may be deduced by using bulk asymptotics in Theorem 2.2 when ζ is inside or on the Szegő curve, or error-function asymptotics when ζ 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 ζ is inside or outside of the Szegő curve.
As already indicated, we will make use of (and develop) the method of approximate full-plane 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 square-root of the Berezin kernel: 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 Subsection 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 full-plane orthogonal polynomials is found in the paper [18], where strong asymptotics with respect to certain special types of potentials is deduced using Riemann-Hilbert 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 Subsection 6.6 below.In [58] it is noted that planar orthogonal polynomials can be characterized as the unique solution to a certain matrix-valued ∂-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 Section 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 Section 2 we consider the Ginibre ensemble and prove Theorem 1.1 and Corollary 1.2.
In Section 3 we provide some necessary background for dealing with more general random normal matrix ensembles.
In Section 4, we state an approximation formula for Kn (z, w) in (1.30) valid when z and w belong to N (U, δ n ).This formula expresses Kn (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 Section 5, we provide a self-contained proof of the main approximation lemma used in Section 4. Our exposition is based on the method in [54], but is easier since (for example) we only require leading order asymptotics.
In Section 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.
Acknowledgement.We want to thank P.J. Forrester for helpful communication.

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 w belongs to the exterior Szegő domain E sz .In addition, we shall state and prove Theorem 2.2 on bulk type asymptotics.
, 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]).
We now define a curve We integrate in (2.2) over the curve connecting ζ to ∞ 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 Re t → +∞ as t → ∞ along the curve; Figure 6 illustrates the point.We find From te 1−t = u we obtain Now write f (u) = (t(u) − 1) −1 and consider the point Then f (A) = (ζ − 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 iθ x where 0 ≤ x ≤ |A|. . (2.5) Inserting here u = u(ζ) = A and using that t(u(ζ)) = ζ and By induction, one shows easily that the higher derivatives have the structure where r j (ζ) is a rational function having a pole of order 2j + 1 at ζ = 1 and no other poles in Ĉ.
On account of (2.5), (2.6), (2.7) and since where rj (z) is a new rational function with pole of order 2j at ζ = 1.
Recalling that ζ = z w and using (2.3) and Lemma 2.1, where the expression in brackets is short for (2.9) 1 − 1 n and each ρ j (ζ) is a rational function with a pole of order 2j at ζ = 1 and no other poles.
We have arrived at the expansion formula (1.4) in the case when ζ = z w is strictly to the right of the curve K.
Next we suppose that ζ = z w is strictly to the left of the curve K. (Thus ζ is either in I or in II or on the common boundary of these domains.) This time we can find a curve of constant argument of u(t) = te 1−t connecting 0 with z, along which |u(t)| is strictly increasing.See Figure 7.We now integrate in (2.2) (using the fact that E n (0) = 1) to write (2.10) 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), (2.11) where the expression in brackets is precisely the same as in (2.9).
Recalling that ζ = z w we obtain, as a consequence of (2.10) and (2.11) that and hence, by (2.1), (2.12) The asymptotic formula in (2.12) has proven for all ζ = z w to the left of the curve K.
We next note that if ζ = z w is in the region III, i.e., if |z we 1−z 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 ζ = z w happens to be precisely on the curve K and ζ ̸ = 1.In this case, we consider nearby points ζ ′ which are either to the left or to the right of K and use a limiting procedure, as ζ ′ → ζ to deduce that the asymptotic formula (1.4) is true in this case as well.(Intuitively, one can picture that for ζ ∈ K we connect ζ either to 0 or to +∞ by first following the curve K until we reach t = 1, and then continue along the real axis until we reach either 0 or +∞.This picture is however not entirely rigorous, since 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 w ∈ C \ (E sz ∪ {1}) we write ρ = |z we 1−z w|.Then ρ ≤ 1 and we have the bulk-asymptotic formula where the implied O-constant is uniform for z w in the complement of any neighbourhood of 1.
Proof.The asymptotic formula in (2.12) applies since ζ = z 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 ).This has been well-known 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 D e , and consider the Berezin measure dµ n,z (w) = B n (z, w) dA(w).We aim to prove that µ n,z converges to the harmonic measure ω z in a Gaussian way.
For this purpose, we fix a sequence (c n ) of positive numbers with nc 2 n → ∞ and nc 3 n → 0 as n → ∞; we can without loss of generality assume that c n < 1 for all n.We then consider points w in the belt N (T, c n ), represented in the form A computation shows that Let μn,z (θ, t) = µ n,z • f n (θ, t) be the pull-back of µ n,z by f n .Also fix θ ∈ T and consider the radial cross-section Since dμ n,z (θ, t) = ϱ n,θ (t) dθdt, it suffices to study asymptotics of the function ϱ n,θ (t).Fixing t we now define an n-dependent point w = f n (θ, t), as in (2.14).Then by Theorem 1.1 |zw| 2n e 2n−2n Re(z w) e 2n Re(z w)−n|z| 2 −n|w| 2 1 After some simplification using that where where the last equality follows by straightforward simplification.

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 Ĉ \ S containing infinity.We also write Γ = ∂U and χ : D e → U for the conformal mapping that satisfies χ(∞) = ∞ and χ ′ (∞) > 0.
Lemma 3.1.Let p be an arbitrary point on Γ.There exists a neighbourhood N of p and a "local Schwarz function", i.e., a holomorphic function S (z) on N \ S, continuous up to N ∩ Γ and satisfying S (z) = z there.
Proof.Without loss of generality set p = 0.
Choosing the neighbourhood N sufficiently small we can write Q(z) = ∞ j,k=0 a j,k z j zk with convergence for all z in N .By polarization we define Here Q is the obstacle function, defined in Subsection 1.2.In N \S, the function ∂ Q is holomorphic and we further have that ∂ Q = ∂Q on N ∩(∂S).Hence G(z, w) is a holomorphic function of z for z ∈ N \S, G(0, 0) = 0 and ∂ w G(z, w)| (0,0) = ∆Q(0) > 0.Moreover, the identity so G(z, z) = 0 for all z ∈ N ∩ S, and, in particular, for all z ∈ N ∩ (∂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 S (z) to the equation G(z, S (z)) = 0, z ∈ N .
Then for z ∈ N \ S we obtain 0 = ∂z G(z, S (z We also find that The proof is immediate from Sakai's regularity theorem in [70], since Γ is a continuum and since a local Schwarz function for U exists near each point of Γ by Lemma 3.1.Similar as for the case τ = 1, the function Qτ is C 1,1 -smooth on C and harmonic on C \ S τ where S τ = S[Q/τ ] is the droplet in potential Q/τ , while Q = Qτ on S τ .(See [61,69].)
Recall that dσ = ∆Q • 1 S dA denotes the equilibrium measure in external potential Q.It is easy to see that σ(S τ ) = τ, and that the restricted measure σ τ defined by By hypothesis, Γ = Γ 1 is everywhere regular (real-analytic).From this and basic facts about Laplacian growth [48,54] we conclude that there are numbers τ 0 < 1 and ϵ > 0 such that Γ τ is everywhere regular whenever τ 0 − ϵ ≤ τ ≤ 1. Indeed τ 0 and ϵ can be chosen so that each potential Q/τ with τ 0 − ϵ ≤ τ ≤ 1 is admissible in the sense of Subsection 1.2.
where the O-constants are uniform in z.In particular there are constants 0 < c 1 ≤ c 2 such that For a proof we refer to [54, Lemma 2. Modifying τ 0 < 1 and ϵ > 0 if necessary, we may assume that V τ is well-defined and harmonic on C \ K where K is a compact subset of Int Γ τ0−ϵ .The set K can be chosen depending only on τ 0 and ϵ and not on the particular τ with τ 0 ≤ τ ≤ 1.
We shall frequently use the following identity: where Q τ is the unique holomorphic function on (This follows since the left and right sides agree on Γ τ and have the same order of growth at infinity.) We turn to a few basic estimates for the function Q − V τ , which we may call "τ -ridge".
Lemma 3.4.Suppose that τ 0 − ϵ ≤ τ ≤ 1 and let p be a point on Γ τ .Then for ℓ ∈ R, where the O-constant can be chosen independent of the point p ∈ Γ τ .
Proof.Using that Q = Qτ on S τ and that Qτ is C 1,1 -smooth, we find that , where ∂ ∂n and ∂ ∂s denote differentiation in the normal and tangential directions, respectively.The result now follows from Taylor's formula.□ Our next lemma is immediate from Lemma 3.4 when z is close to Γ τ 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 τ 0 − ϵ ≤ τ ≤ 1 and that z is in the complement C \ K, where K ⊂ Int Γ τ0−ϵ is defined above.Then with δ τ (z) = dist(z, Γ τ ) 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 τ, τ ′ are in the interval [τ 0 − ε, 1].For a given point z ∈ Γ τ ′ let p ∈ Γ τ be the point closest to z.Then In particular, if τ 0 < 1 and ϵ > 0 are chosen close enough to 1 and 0 respectively, then there are constants c 1 and c 2 independent of τ , τ ′ , 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 tail-kernel 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 well-known in the literature.Lemma 3.7.Let W = P • e − 1 2 nQ be a weighted polynomial where j = deg P ≤ n.Put τ (j) = j/n and suppose τ (j) ≤ τ where τ satisfies 0 < τ ≤ 1.There is then a constant C depending only on Q such that for all z ∈ C, Proof.Let M τ be the maximum of W over S τ .We shall first prove that To this end we may assume that M τ = 1.Consider the function which is subharmonic on C and satisfies s ≤ Q on Γ τ .Moreover, s(w) ≤ 2τ log |w| + O(1) as w → ∞.Hence by the strong maximum principle we have s ≤ Qτ on C, proving (3.6).We next observe that there is a constant C independent of τ 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.□ We shall need to compare obstacle functions Qτ (z) for different choices of parameter τ .It is convenient to note the following two lemmas.Lemma 3.8.Suppose that τ 0 ≤ τ ≤ τ ′ ≤ 1.Then there is a constant c > 0 depending only on τ 0 and Q such that Then H is harmonic on U (including infinity) and has boundary values For a given z ∈ Γ τ ′ we let p ∈ Γ τ be the closest point and write z = p + ℓ • n τ (p).Then |ℓ| ≍ τ ′ − τ by Lemma 3.3, and by Lemma 3.4 Increasing τ 0 < 1 a little if necessary, we obtain H ≥ c(τ ′ − τ ) 2 everywhere on Γ τ ′ where c > 0 is a constant depending on τ 0 and Q.By the maximum principle, the inequality H ≥ c(τ ′ − τ ) 2 persists on U τ ′ .□ Lemma 3.9.Let W = P • e − 1 2 nQ be a weighted polynomial where j = deg P ≤ n with ∥W ∥ = 1.Suppose also that τ (j) ≤ τ where τ 0 ≤ τ ≤ τ ′ ≤ 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 □ Finally, we arrive at following estimate, which will be used to discard lower order terms in the tail-kernel approximation in the succeeding section.Lemma 3.10.Let Suppose that τ (j) ≤ θ n and let W j,n (z) be the j:th weighted orthonormal polynomial in the subspace W n ⊂ L 2 (C).There are then constants C and c > 0 depending only on Q such that Proof.We may assume that τ (j) ≥ τ 0 where τ 0 < 1 is as close to 1 as we please.We apply Lemma 3.9 with Applying Lemma 3.9, we find that (with a new C) we finish the proof by choosing c > 0 somewhat smaller.□

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 Subsection 1.2.where W j,n = P j,n • e − 1 2 nQ , is the j:th weighted orthogonal polynomial, i.e., P j,n has degree j and positive leading coefficient.The numbers θ n and δ n are defined by (4.2) 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 ∞.Lemma 4.1.("Main approximation lemma") Suppose that z, w ∈ N (U, δ n ) and let β be any fixed number with 0 < β < 1  4 .Then with τ (j) = 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 Section 5.
We now turn to the proofs of our main results (Theorems 1.3 and 1.5).Towards this end (using notation such as Q = Q 1 and ϕ = ϕ 1 ) we rewrite (4.3) as where we used the notation Our main task at hand is to estimate the sum Sn (z, w).
Remark.In going from (4.3) to (4.4) we used the facts that where the O-constants are uniform for z in U τ and (say) nθ n ≤ τ ≤ n.This follows by an application of the maximum principle, using that the functions are holomorphic on Ĉ \ K and that relevant estimates are clear on the boundary curve Γ τ .
We have the following main lemma, in which we fix a small number η > 0. Lemma 4.2.Suppose that z, w ∈ N (U, δ n ) and that |ϕ(z)ϕ(w) − 1| ≥ η.Then there is a positive constant N such that The constant N as well as the O-constant can be chosen depending only on the parameters η and M , and on the potential Q.
Taken together with (4.4), the lemma gives a convenient approximation formula for the tail Kn (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 Subsection 4.4.

4.2.
Preparation for the proof of Lemma 4.2.For τ close to 1 we introduce the following holomorphic function on Ĉ \ K, Notice that F τ (∞) > 0 and that (4.6) can be written For the purpose of estimating Sn (z, w) we write We also denote m = ⌊nθ n ⌋, the integer part of nθ 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 ∈ C \ K, (4.9) We shall find below that a n−1 → 1 and a m → 0 quickly as n → ∞.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, δ n ) and that |ϕ(z)ϕ(w) − 1| ≥ η, and we write τ (j) = j n .We begin with the following lemma.Lemma 4.4.Let h(z) be the unique holomorphic function in a neighbourhood of U which satisfies the boundary condition and the normalization Im h(∞) = 0. Then for all z, w in a neighbourhood of U and all j such that τ 0 ≤ τ (j) ≤ 1 we have as n → ∞ where b 3 (z) is a holomorphic function in a neighbourhood of U .
Before proving the lemma, we note that the harmonic function Re h(z) defined by the boundary condition (4.10) is strictly negative in a neighbourhood of U by the maximum principle.
Hence Lemma 4.4 implies the following result.
Corollary 4.5.By slightly increasing the compact set K ⊂ C \ U if necessary, we can ensure that for all z, w and there is a constant s > 0 such that (with m = nθ 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 ∈ Ĉ \ K and real τ near 1 we consider the function where we use the principal determination of the logarithm, i.e., Im P (τ, ∞) = 0.It is clear that We now consider the Taylor expansion in τ , about τ = 1, (4.12) which we write as whence by the asymptotics in Lemma 3.6, we have as τ → 1, Comparing with (4.12) we infer that the holomorphic functions b 1 and b 2 on Ĉ \ K satisfy Re b 1 = 0 on Γ 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 a j (z, w) = exp n(P (τ (j), z) + P (τ (j), w)) and refer to (4.15).□ At this point, it is convenient to switch notation and write where then 1 ≤ k ≤ √ n log n.We will denote and assume that this is an integer.We will also write (4.17) The following lemma is a direct consequence of Lemma 4.4.
Lemma 4.6.For n − µ ≤ j ≤ n − 1 we have the asymptotic (as n → ∞) Proof.This is immediate on writing n and inserting the asymptotics in Lemma 4.4; details are left for the reader.□ We are now ready to give our proof of Lemma 4.2.
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 (4.20) with some constant N .Using Lemma 4.6, it is seen that From (4.18) we have the lower bound Also, since r ≥ 1 − Cδ n we have We now show that σ n,1 and σ n,2 are negligible as n → ∞.
To treat the case of σ n,1 we write a k = e ndε k = e dk , so that k . Using Making use of a Riemann sum and the substitution t = √ nε, we get where we put C = CM/2.
In the case when r is "large" in the sense that r ≥ r 0 > 1, we can do better.Indeed since Re d = − log r, the partial sums B k | ≲ r −1 where the implied constant depends on r 0 .The method of estimation above thus gives The term σ n,2 can be handled similarly: we introduce the notation a One deduces without difficulty that |a A straightforward adaptation of our above estimates for σ n,1 now leads to Moreover, in the case when r ≥ r 0 > 1 we obtain the improved estimate |σ n,2 | ≲ 1/(r √ 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 α = − Re c > 0 we thus consider the following four terms: By a Riemann sum approximation and the estimate (4.19) we find σn,ν ∼ n ν Since (for 0 ≤ ν ≤ 3) It is also easy to verify that for r ≥ r 0 > 1 we have σn,ν ≲ 1/(r √ 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) ) Combining these estimates, we find in all cases that r n .We have reached the desired bound (4.20) with N = C 2 /α 2 , and our proof of Lemma 4.2 is complete.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 Kn (z, w).
To prove this we first note that Lemma 4.2 implies that the size of the tail-kernel is To estimate lower order terms, corresponding to j with τ (j) ≤ θ n , we recall Lemma 3.10 that there is a number c ′ > 0 such that for all z ∈ N (U, Using a similar estimate for W j,n (w) and picking any c > 0 with c < c ′ , we conclude the estimate Since Q = V on U , we obtain from (4.27) and (4.29) that K n (z, w) = Kn (z, w) )) in the case when both z and w are in U .However, since z and w are allowed to vary in the δ 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 ∈ N (U, δ n ), Proof.Since Q = V on U , we can assume that z ∈ S. Then Q(z) = Q(z).Let p ∈ Γ be the closest point and write z = p + ℓn 1 (p) where |ℓ| ≲ δ n by Lemma 3.3.The Taylor expansion in Lemma 3.4 now shows that (Q − V )(z) = 2∆Q(p)ℓ 2 + O(ℓ 3 ), finishing the proof of the claim.□ Combining (4.29) with (4.30) we conclude that if z, w ∈ N (U, δ 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 Kn (z, w) has the desired asymptotic when the points z, w belong to N (U, δ n ) and |ϕ(z)ϕ(w) − 1| ≥ η.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 ∈ U and recall that We express points w in C (in some fixed neighbourhood of Γ) as w = p+ℓ•n 1 (p) where p = p(w) is a point on Γ, n 1 (p) is the unit normal to Γ pointing outwards from S and ℓ is a real parameter.
Given a point p ∈ Γ we also recall the Gaussian probability measure γ p,n on the real line, (4.32) Denote by ω z = ω z,U the harmonic measure of U evaluated at z and consider the measure dμ n,z = dω z (p) dγ n,p (ℓ); writing dω z (p) = P z (p) |dp|, we have By Theorem 1.3 we have, for fixed z ∈ U and any w ∈ N (U, δ n ), Recalling that |ϕ(w)| 2n e n Re Q(w) = e nV (w) , we obtain We next recall that (by Lemma 3.5), the factor e −n(Q−V )(w) is negligible when dist(w, Γ) ≥ δ n , so we can focus on the asymptotics of (4.33) in the δ n -neighbourhood N (Γ, δ n ).
Near the curve Γ, Lemma 3.4 gives where O-constant is independent of the point p ∈ Γ.
(The convergence o(1) → 0 holds in the uniform sense of densities on the sets where |t| ≤ 2M √ log log n.)At this point we notice that the measure is precisely the harmonic measure for D e evaluated at the point ϕ(z) ∈ D e (cf.[44]).
Pulling back the left and right hand sides in (4.37) by the inverse f −1 n (p, ℓ) = (ϕ(p), 2 √ nℓ) and using conformal invariance of the harmonic measure, we infer that the measure µ n,z satisfies and that the uniform convergence on the level of densities asserted in (1.29) holds.(The factor 1 π in the left hand side comes from our normalization of the area measure dA.) q.e.d.

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 } n−1 j=0 to be the orthonormal basis for the weighted polynomial subspace W n of L 2 with W j,n = P j,n • e − 1 2 nQ , where the polynomial P j,n has degree j and positive leading coefficient.The tail kernel Kn (z, w) is then given by (5.1) Kn (z, w) As always, we write where U is the component of Ĉ \ S containing ∞.

5.1.
Reduction of the problem.Fix numbers τ 0 < 1, ϵ > 0 and a compact subset K ⊂ Int Γ τ0−ϵ with the properties in Subsection 3.2.Also fix j and n such that τ 0 ≤ τ (j) ≤ 1, where (as always) τ (j) = j/n.Following [54] we define an approximation of W j,n (z) on C \ K by (5.2) where Here H τ and Q τ are bounded holomorphic functions on Ĉ \ K with Re H τ = log √ ∆Q and Re Q τ = Q on Γ τ ; ϕ τ is the univalent extension to Ĉ \ K of the normalized conformal map U τ → D e .Lemma 5.1.("Main approximation formula") The number τ 0 < 1 may be chosen so that if τ 0 ≤ τ (j) ≤ 1 and if β is any number in the range 0 < β < 1  4 then as n → ∞, It is clear that Lemma 5.1 implies Lemma 4.1 on asymptotics for the tail kernel Kn (z, w).The rest of this section is devoted to a proof of Lemma 5.1.

Foliation flow.
One of the key ideas in [54] is to introduce a set of "flow coordinates" to facilitate computations.
Note that ψ t continues analytically across T and obeys the basic relation Indeed, our definitions have been set up so that, for all large n, With τ (j) = j/n, we define a neighbourhood D j,n of T by (5.7) D j,n = −2εn≤t≤2εn ψ τ (j),t (T).
The inverse image ϕ −1 τ (j) (D j,n ) plays the role of an "essential support" for W j,n and W ♯ j,n .
5.3.Approximation scheme.In the following we fix j and n with τ 0 ≤ τ (j) ≤ 1, where τ (j) = j/n.We shall extend W ♯ j,n to a smooth function on C by a straightforward cut-off procedure.
It is convenient to modify the compact set K ⊂ Int Γ τ0−ϵ so that ϕ τ (j) maps Ĉ \ K biholomorphically onto some exterior disc D e (ρ 0 − δ) where ρ 0 < 1 and δ > 0. (Then K = K(j) may slightly vary with j, but it will be harmless to suppress the j-dependence in our notation.) Next we fix a smooth function χ 0 such that χ 0 = 0 on K and χ 0 = 1 on ϕ −1 τ (j) (D e (ρ 0 )) and define (5.8) (It is understood that W ♯ j,n = 0 on K.) The following properties of the function W ♯ j,n are key for what follows: (1) W ♯ j,n is asymptotically normalized: is approximately orthogonal to lower order terms: 2 nQ ∈ W n with degree P < j.

5.4.
Positioning and the isometry property.Continuing in the spirit of [54], we define the "positioning operator" Λ j,n by Also define a function ("τ (j)-ridge") by where V τ is the harmonic continuation of Qτ Uτ inwards across Γ τ .

□
It follows that if f is an integrable function on D j,n then (5.14) so by Lemma 5.3, (5.15) j) and using that R τ (j) • ψ t = t 2 on T, we now see that and by Lemma 5.2, the last term on the right is O( √ ne −c log 2 n ) for a suitable constant c > 0. We have shown the approximate normalization property (1), i.e., we have shown: 5.6.Approximate orthogonality.We now prove property (2) of the quasipolynomials.Given a positive integer k, it is convenient to write W k,n for the space of weighted polynomials W = P • e − 1 2 nQ where P has degree at most k, equipped with the usual L 2 -norm.
By the Cauchy-Schwarz inequality and Lemma 5.2 we conclude that Hence it suffices to estimate the integral 16) where h = q/f j,n is holomorphic of D e (ρ 0 ) and vanishes at infinity (since f j,n (∞) > 0).
By (5.15), by the mean-value property of holomorphic function, so we obtain the estimate Since 1/f j,n is bounded on Dn we see that Using the Cauchy-Schwarz inequality the right hand side is estimated by |q| 2 e −nR τ (j) dA) where we used the isometry property (5.9) to deduce the equality.□ 5.7.Pointwise estimates.We wish to show that when τ (j) is close to 1, then W j,n is "pointwise close" to W ♯ j,n near the curve Γ τ (j) .
Lemma 5.6.There are constants C and n 0 such that for all n ≥ n 0 and all j with τ 0 ≤ τ (j) ≤ 1 we have ∥W j,n − W ♯ j,n ∥ ≤ Cε n .
Proof.Let u 0 be the norm-minimal solution in L 2 (e −nQ , dA) to the following ∂-problem: A standard estimate found in [57,Section 4.2] shows that there is a constant C such that Since ∂χ 0 = 0 on U τ0−ϵ , Lemma 5.2 implies that there is a constant c > 0 such that |F j,n | 2 e −nQ ≤ e −cn on the support of ∂χ 0 .Thus (5.17) ∥u 0 ∥ L 2 (e −nQ ) ≤ Ce −cn with (new) positive constants C and c.We correct F j,n • χ 0 to a polynomial Pj,n of exact degree j by setting ( Pj,n is then an entire function of exact order of growth O(z j ) as z → ∞, since |F j,n (z)| ≍ |z| j and |u 0 (z)| ≲ |z| j−1 as z → ∞, so indeed Pj,n is a polynomial of exact degree j.)It follows from (5.17) that Similarly, the approximate orthogonality in Lemma 5.5 implies (with the estimate (5.18)) that 2 ) by (5.21), and so (5.20) and (5.19).
Moreover, since W * j,n ∈ W j,n ⊖ W j−1,n = 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 ∥W j,n ∥ = 1 we then have c j , and the proof of the lemma is complete.□ Following a well-known circle of ideas we shall now turn the L 2 -estimate in Lemma 5.6 into a pointwise one.

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 one-point function.
An advantage of the loop equation point of view is that it continues to hold in a context of βensembles, 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 n-expansions can be found in the papers [5,11,12,15,22,30,31,39,53,59,78] 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 1-point function and R n,k (w 1 , . . ., w k ) = det(K n (w i , w j )) k×k for the k-point function of the determinantal Coulomb gas process {z j } n 1 associated with Q.By Theorem 1.5 we know that if z is in the exterior domain U , then the Berezin measure µ n,z obeys the asymptotic (6.1) where ω z and γ n,p denote certain harmonic and Gaussian measures, respectively.As we shall see (whether or not z is in the exterior) µ n,z is an exact solution to the loop equation (We remind that µ(f ) is short for f dµ.) The relation (6.2) is not the "usual" form of the two-dimensional loop equation (e.g.[11]), but rather a kind of infinitesimal variant; for completeness we include a derivation of it below.6.2.β-ensembles.Given a large n and a configuration {z j } n 1 we consider the Hamiltonian The Boltzmann-Gibbs law in external potential Q and inverse temperature β is the following probability law on C n , (6.3) Suppose that {z j } n 1 is picked randomly with respect to (6.3).For fixed k ≤ n we denote by R β n,k the k-point function, i.e., the unique (continuous) function on C k obeying Finally, we introduce the Berezin kernel B β n (z, w) and the Berezin measure µ β n,z by In the following we denote by R β n = R β n,1 the 1-point function.
Proof.Let Λ ⊂ C be an open set such that Q is C 2 -smooth in a neighbourhood of the closure cl Λ. Fix a point z ∈ Λ and a smooth real-valued function ψ supported in Λ.Given a random sample {z j } n 1 , we can for each j view the number ψ(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, ∂ j is short for ∂/∂z j .) Now observe that for each j Hence a summation in j gives We have shown that (6.4) 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 1-point function and an integration by parts we have dA(w).
Since the identity (6.4) holds for every test-function ψ, we obtain the pointwise identity for all z ∈ Λ, dA(w) = 0. (6.5) (First we obtain the identity in the sense of distributions on Λ, then everywhere, since the functions involved are smooth.) Dividing through by βR β n,1 we find 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) ≤ K n (w, w) and K n (w, w) converges to zero uniformly for w outside of any given neighbourhood of (Int Γ) ∪ Γ, 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 \ {∞}; the extension to more general admissible potentials may be left to the reader.
We shall use Theorem 1.3, which implies that for z ∈ U , Beyond the Ginibre ensemble, it is not clear from our above results that there is a similar large n-expansion.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 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 ρ j (z, z), which are proportional to b j (z, z)/b 0 (z, z).□ By inserting the ansatz (6.10) in the loop equation, we get a feed-back relation for the correction terms ρ j (z, z).A deeper analysis of this structure is beyond the scope of our present investigation.6.5.Back to β-ensembles.We finish with a few words about β-ensembles.In the case when z ∈ U , it is known due to the localization theorem in [5]  (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 ∂ ∂ z (µ β n,z (k z )) = O(1) when z ∈ U .Assuming that this is the case, and comparing O(n)-terms in Ward's identity we find "heuristically" the approximation (6.13) ∆ log R β n (z) = −nβ∆Q(z) + • • • , where the dots represent terms of lower order in n.
When z ∈ U is close to the boundary ∂U , (6.13) is consistent with predictions found in [30], and also with the localization theorem in [5].
The papers [15,30,31] and the references there provide more information about the problem of finding asymptotics for the 1-point function R β n when β > 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 Γ = ∂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 S at each point p ∈ Γ can be established precisely as in the proof of Lemma 3.1.It follows by Sakai's main result in [70] that C \ U has finitely many components K 1 , . . ., K d , and that the normalized Riemann maps χ l : D e → Ĉ \ K l can be continued analytically across T for l = 1, . . ., d.We put Γ l = χ l (T) = ∂K l and assume that χ ′ l ̸ = 0 on T for l = 1, . . ., n.Then each Γ l is an analytic, non-singular Jordan curve.For a basic model case we consider the disconnected lemniscate droplet, defined by the potential  The following problem presents itself: to find the asymptotics of K n (z, w) when z and w belong to different boundary components of ∂S.Since U is not simply connected, there is no longer a Riemann map ϕ, 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 Riemann-Hilbert techniques following the works [18,19,29,63].This was used to generate Figure 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 gap-probabilities.(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 lemniscate-type singularities which are found at boundary points where ∆Q = 0).More background on singular points can be found e.g. in [13] and the references there.

Figure 2 .
Figure 2. Plot of the Berezin kernel for the Ginibre ensemble, w → B n (z, w) for n = 20 and z = 2.

1. 3 . 1 . 3 . 1 .Theorem 1 . 3 .
Asymptotic results for admissible potentials.In the following, Q denotes an admissible potential in the sense of Subsection 1.2.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, δ n ) denotes the δ n -neighbourhood of the exterior set U , cf. (1.21).)Fix constants η and β with η > 0 and 0

Figure 4 .
Figure 4.The Berezin kernel w → B n (z, w) where z = 2 and n = 20.Here Q is the elliptic Ginibre potential Q(w) = u 2 + 3v 2 where w = u + iv.The droplet S is the elliptic disc 1 2 u 2 + 6v 2 ≤ 1, so z belongs to the exterior component U and the emergent Gaussian approximation of harmonic measure is clearly visible.
Stirling's formula (see Lemma 2.1 below) we can now conclude Theorem 1.1 in the case | arg(z w − 1)| < 3π 4 .Conversely, we can use Theorem 1.1 to conclude the following generalized version of Tricomi's expansion.

Figure 5 .
Figure 5. Regions and curves used in the proof of Theorem 1.1.

Lemma 2 . 1 .
There are numbers b k starting with b 0 = 1 and b K and three regions I, II, III using the function u(ζ) = ζ e 1−ζ , depicted in Figure 5.The regions I (bounded) and II (unbounded) are defined to be the connected components of the set {|u(ζ)| < 1}.(Note that I = Int γ sz is the domain interior to the Szegő curve (1.3).)We also define III := {|u(ζ)| > 1}.Note that u(ζ) has a critical point at ζ = 1.We define the curve K to be the portion of the level curve Im u(ζ) = 0 which intersects the real axis at right angles at ζ = 1.We assume that ζ ̸ = 1 and divide in two cases according to which ζ is to the left or to the right of the curve K. (The case when ζ is exactly on K \ {1} will be handled easily afterwards.)First assume that ζ is strictly to the right of K. (So ζ is either in region II or in region III or on the common boundary of those regions.)

Figure 6 .
Figure 6.Curves of constant argument connecting ζ with +∞ when ζ is to the right of K.The first picture shows a curve where u(t) is real; the second picture has the argument of u(t) equal to π 4 + 2πk.

Figure 7 .
Figure 7. Curves of constant argument connecting 0 with ζ when ζ is to the left of K.The first picture shows a curve where u(t) is real; the second picture has the argument of u(t) equal to π 4 + 2πk.
z)) = 0 for z ∈ N ∩ (∂S), so S (z) = z at such points.□ Theorem 3.2.The conformal map χ : D e → U extends analytically across T to an analytic function on a neighbourhood of the closure of D e .As a consequence Γ = χ(T) is a finite union of real analytic arcs and possibly finitely many singular points, which are either cusps (corresponding to points p = χ(z) with z ∈ T and χ ′ (z) = 0) or double points (p = χ(z 1 ) = χ(z 2 ) where z 1 , z 2 ∈ T and z 1 ̸ = z 2 ).
For fixed τ ∈ (0, 1] we let Qτ be the obstacle function which grows likeQτ (z) = 2τ log |z| + O(1), z → ∞.By this we mean that Qτ (z) is the supremum of s(z) where s runs through the class SH τ (Q) of subharmonic functions s on C which satisfy s ≤ Q on C and s(w) ≤ 2τ log |w| + O(1) as w → ∞.

(4. 21 )
Here A, B, c are certain complex numbers depending on z and w; the important fact is that Re c < 0. To analyze the right hand side in (4.21), we set d = − log r + iϑ and introduce the notation

6 . 4 .Theorem 6 . 2 .
z − w dA(w) − µ β n,z (k z ) = 0.Taking ∂-derivatives with respect to z in the last identity, we finish the proof.□ On various asymptotic relations.Now set β = 1.We have the following theorem on the Cauchy transform µ n,z (k z ) of the Berezin measure µ n,z .(The symbol k z denotes the Cauchy kernel k z (w) = (z − w) −1 , and ω z denotes the harmonic measure of U evaluated at a point z ∈ U .)With ϕ : U → D e the normalized conformal map, we have the identity(6.6)lim n→∞ µ n,z (k z ) = ω z (k z ) = ∂ ∂z log(|ϕ(z)| 2 − 1) + H(z), (z ∈ U )where H(z) is a holomorphic function in U with H(z) = O(z −2 ) as z → ∞.