The Random Normal Matrix Model: Insertion of a Point Charge

In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for log|pn(ζ)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\log |p_{n}(\zeta )|$\end{document} where pn is the characteristic polynomial of an n:th order random normal matrix.


Introduction and main results
Insertion operations appear frequently in Coulomb gas theory, conformal field theory, and allied topics. In this article we study planar Coulomb gas ensembles {ζ j } n 1 in the determinantal case, under the influence of an external confining potential Q and we investigate the effect of inserting a point charge at a point p in the bulk of the droplet created by the external field. The inserted charge is assumed to be of the same order of magnitude as the individual charges ζ j .
From a field-theoretical point of view, this kind of insertion operation is quite natural, for example it can be directly related with the statistics of the random logarithmic potential Y n (p) = n 1 log |p − ζ j |. The investigation of random fields such as Y n and their asymptotic properties as n → ∞ is currently an active area; see for instance [16,17,20,21,26,27,28,29,31,33,34,35,36,38,41,42] for related works. Our main contribution in this direction is a central limit theorem for Y n (p) which is valid for the class of algebraic potentials (of the form Q(ζ) = τ 0 |ζ − p| 2λ + P (ζ) near the droplet, where τ 0 , λ > 0, P is a harmonic polynomial, and (say) Q ≡ +∞ near ∞).
Our exposition depends on a knowledge of the microscopic scaling limit near the inserted charge. Assuming that the potential is algebraic and using theory for rescaled Ward identities in the spirit of the papers [3,5,6,9], we are in fact able to find an explicit microscopic correlation kernel, depending on three parameters: τ 0 , λ, and the strength of the inserted charge. This part of our work is connected with the theory of certain Fock-Sobolev spaces of entire functions, and might be of some independent interest.
General notation. If g is a function, thenḡ(z) means the complex-conjugate of g(z). A function h(ζ, η) is Hermitian if h(η, ζ) =h(ζ, η) and Hermitian-analytic (entire) if it is moreover analytic (entire) in ζ andη. A cocycle is a function c(ζ, η) = g(ζ)ḡ(η) where g is a continuous unimodular function. C * = C \ {0} is the punctured plane,Ĉ = C ∪ {∞} is the Riemann sphere, D(a; r) is the open disk with center a, radius r, and Pol(n) is the set of analytic polynomials of degree at most n − 1. If K ⊂ C is compact then the "polynomially convex hull" Pc K is the complement of the unbounded component of C \ K. (Cf. e.g. [40, p. 53]. ) We write ∆ = ∂∂ = 1 4 ( ∂ 2 ∂x 2 + ∂ 2 ∂y 2 ) (usual Laplacian divided by 4) and dA = 1 π dxdy (area measure divided by π). Also, we sometimes write µ(f ) for f dµ. In what follows, we will assume some basic familiarity with reproducing kernels and Bergman spaces; see [18] and [10] for introductions to these theories.
We start by fixing a suitable lower semicontinuous function (external potential) Q : C → R ∪ {+∞} of sufficient increase near ∞, For a Borel measure µ on C we associate the weighted logarithmic energy, The minimizer σ = σ[Q] of this energy, among compactly supported Borel probability measures µ, is the equilibrium measure in external potential Q. The support S = S[Q] := supp σ is called the droplet. Assuming some smoothness of Q in a neighbourhood of the droplet we ensure that S is compact and σ is a probability measure taking the form dσ = ∆Q · 1 S dA. In particular we have ∆Q ≥ 0 on S.
(We refer to [40] as a source for these results; see especially Theorem I.1.3, Theorem II.1.3. In our case Q will not be everywhere smooth near the droplet, but merely satisfy a Hölder condition at the origin, so the remarks about nonsmooth potentials on page 138 are also relevant.) In the following, we will need some further regularity of the droplet. To guarantee this, by means of Sakai's theory, it suffices, besides the growth (1.1), to assume that Q be real-analytic in some neighbourhood of the boundary ∂S. We refer to the discussion in [7, Sect. 1] for further details.
To simplify the discussion, we will assume that Q be real-analytic wherever Q < +∞, except possibly at the origin, where we merely assume that the limit (1.2) λ − 1 := lim ζ→0 log ∆Q(ζ) 2 log |ζ| exists and exceeds −1, i.e., we require that λ > 0. This implies that Q is Hölder continuous at the origin; see (1.5) below. Let us fix a ∈ Int S, say a = 0. Also fix a number c > −1, a positive integer n, and a suitable, smooth, real-valued function u. Given this, we define an n-dependent potential (1.3) V n (ζ) = Q(ζ) + 2c n log 1 |ζ| − 1 n u(ζ). We refer to Q as the underlying potential -this is what determines the global properties of the droplet. The logarithmic term in (1.3) has an effect on the microscopic distribution of particles near 0 and near the boundary of the droplet. More precisely, it corresponds to a point charge of strength c at the origin, and at the same time, a harmonic measure on the boundary of the droplet with opposite sign. This effect can be understood as a balayage operation, sweeping of a measure on a domain (here, a point mass in the droplet) to the boundary. The extra freedom afforded by the term u/n is used to accommodate different kinds of microscopic behaviour near the boundary.
We now define ensembles. Given V n of the form (1.3), we consider a random system {ζ j } n 1 ⊂ C of n identical point charges under the influence of the external field nV n . The system is picked at random with respect to the Boltzmann-Gibbs probability law (1.4) dP n = 1 Zn e −Hn dA n , H n := j =k where Z n = C n e −Hn dA n is the partition function and dA n = dA ⊗n is suitably normalized Lebesgue measure on C n .
(The system {ζ j } n 1 represents the eigenvalues of n × n random normal matrices equipped with the probability measure proportional to e −ntr Vn(M ) dM . Here, tr is the trace and dM is the measure on the space of normal matrices induced by the Euclidean metric on C n 2 . See e.g. [14] and [19] for details.) Recalling that 0 ∈ Int S we shall study the distribution of rescaled system z j = r −1 n · ζ j , r n := n −1/2λ , where λ > 0 is given by (1.2). The number r n = n −1/2λ might be called a microscopic scale. (This scale is chosen so that σ(D(0, r n )) = Cn −1 + · · · for some positive constant C.) To analyze the large n behaviour of {z j } n 1 , we shall use the canonical decomposition of Q about the origin, by which we mean (1.5) Q(ζ) = Q 0 (ζ) + h(ζ) + O(|ζ| 2λ+ ), where Q 0 is a non-negative, homogeneous function of degree 2λ, and where h is a harmonic polynomial of degree at most 2λ and is a positive number. In this article, we assume that the potential has the above canonical decomposition and that the O-term remains small after taking Laplacians: If Q is real analytic at the origin, then we surely obtain the canonical decomposition by Taylor series expansion. We also assume that Q 0 and ∆Q 0 are positive definite, i.e., Q 0 > 0 and ∆Q 0 > 0 when z = 0. We refer to the homogeneous function Q 0 as the dominant part of Q. Typically, we shall find that the rescaled ensemble depends in a "universal" way on Q 0 and c.
Remark. Subtracting an n-dependent constant from Q does not change the problem, so we can assume Q(0) = h(0) = 0. We can likewise assume that u(0) = 0. In the following, except when otherwise is explicitly stated, we assume that these normalizations are made.
Definition. We say that the point p = 0 is a regular point if c = 0 and λ = 1; otherwise it is singular of type (λ, c). If c = 0 and λ = 1 we speak of a conical singularity. We have a bulk singularity if c = 0 and λ = 1, and a combined singularity if c = 0 and λ = 1.
We now recall some terminology with respect to the system {ζ j } n 1 and its rescaled counterpart {z j } n 1 . As a general rule, we designate non-rescaled objects by boldface symbols R, K, L, etc., while rescaled objects are written in italics, R, K, L, etc.
For a subset D ⊂ C we consider the random variable N D = #{j : ζ j ∈ D}, and we define, for distinct η 1 , . . . , η p ∈ C, the intensity p-point function The most basic intensity function is the 1-point function R n (ζ) := R n,1 (ζ).
As is well-known (see e.g. the computation in [40, Section IV.7.2]), the function R n,p can be expressed as a p×p determinant R n,p (ζ 1 , · · · , ζ p ) = det (K n (ζ i , ζ j )) p i,j=1 , where K n is a Hermitian function called a correlation kernel of the process. Indeed, we can take K n as where p n,j is the orthonormal polynomial of degree j with respect to the measure e −nVn dA. We shall always use the symbol K n to denote this canonical correlation kernel.
The rescaled system {z j } n 1 has p-point function R n,p (z 1 , . . . , z p ) = det(K n (z i , z j )) where the kernel K n is given by (1.6) K n (z, w) = r 2 n K n (ζ, η), (z = r −1 n ζ, w = r −1 n η). We now define function spaces. We will employ a function V 0 , which we call the microscopic potential at 0, whose definition is Here, Q 0 is the dominant part of Q in (1.5). We also define a corresponding measure µ 0 by dµ 0 = e −V0 dA, and write L 2 a (µ 0 ) for the Bergman space ("generalized Fock-Sobolev space") of all entire functions f of finite norm, f L 2 (µ0) = ( C |f | 2 dµ 0 ) 1/2 . The Bergman kernel of this space will be denoted by L 0 (z, w).
A straightforward computation of the orthonormal polynomials p n,j shows that whence R n (z) converges to the limit .
For example, when τ 0 = λ = c = 1 we obtain the well-known one-point function In the present case V 0 , the microscopic potential, is just V 0 (z) = τ 0 |z| 2λ − 2c log |z|, and µ 0 = e −V0 dA. It is easy to compute the Bergman kernel L 0 of the space L 2 a (µ 0 ). Indeed, using that L 0 (z, w) = ∞ 0 e j (z)ē j (w) where e j are the orthonormal polynomials with respect to µ 0 , one finds Comparing (1.8) and (1.10) we find the basic relation R(z) = L 0 (z, z)e −V0(z) between the microscopic density and the Bergman-kernel in Fock-Sobolev space. We shall see that this relation remains true in a much more general situation.

1.2.
Main results. We now state our two principal results, which concern singular points with a certain "rotational symmetry", which holds up to a harmonic polynomial.
Our first main result asserts that the kernel (1.10) appears universally at this kind of singular point. Theorem 1.1. Suppose that Q and V n are of the form (1.11) and that Q 0 is given by Q 0 (z) = τ 0 |z| 2λ .
If S is connected and if the outer boundary ∂ Pc S is everywhere regular, then the point-processes {z j } n 1 converge as n → ∞ to a unique determinantal point field with correlation kernel K(z, w) = L 0 (z, w)e −V0(z)/2−V0(w)/2 where L 0 is explicitly given as the Mittag-Leffler kernel (1.10).
Remark. The topological assumptions on S in Theorem 1.1 are made for convenience, in order to apply the main result on orthogonal polynomials from [25].
Our next result is a central limit theorem for the normalized Coulomb potential generated by the eigenvalues of a random normal matrix ensemble with respect to an algebraic underlying potential, namely a potential of the special form (near the droplet) where P is a harmonic polynomial. Correspondingly we define the insertion potential This definition requires some extra care, since it may well happen that the growth condition (1.1) fails (if the degree of P is large). The standard way to circumvent this problem is by considering local droplets as in e.g. the papers [19,37], i.e., we redefine Q to be +∞ outside some large compact set.
To be precise, given any (non-polar) compact set Σ ⊂ C, we may redefine Q to be +∞ outside of Σ. The redefined potential has a well-defined droplet S = S Q,Σ . In the following we shall fix a compact set Σ large enough that S ⊂ Int Σ, and then redefine the potential Q in (1.12) as +∞ outside of Σ. (The particular choice of such a set Σ will be immaterial.) We shall always adopt this convention about algebraic potentials in the sequel.
Denote by {ζ j } n 1 a random sample picked with respect to (1.4) with Q in (1.12) and write (ζ) = log |ζ|. It is natural to define a random variable trace n by We shall study the fluctuation about the mean, fluct n := trace n − E n trace n . When 0 is a regular point, it is expected that fluct n be in some sense a good approximation to the Gaussian free field evaluated at 0 provided that n is large, see e.g. the concluding remarks in [3] or [28,Appendix 6].
A problem with making this rigorous is that the variance of the fluctuations grows logarithmically in n. There are various ways one could try to circumvent this difficulty; our approach here is to consider the "normalized" fluctuations X n = 2 √ log n (trace n − E n trace n ). Theorem 1.2. Suppose that Q is algebraic, of the form (1.12). Let {ζ j } be picked randomly with respect to V n in (1.13). Suppose also that the conditions of Theorem 1.1 are satisfied. Then X n converges in distribution to the normal distribution with mean 0, variance 1/λ.
We also note that the distribution of random variables has been well-studied when f is, for example, a characteristic function. In fact, Soshnikov in [41, Theorem 1] has shown thatX n (f ) converges in distribution to the standard normal for a large class of bounded test functions f such that the variance Var n (trace n f ) → ∞ as n → ∞. Theorem 1.2 gives a similar result for the random variablesX n (f ) in the case where f = is unbounded. Indeed, our result immediately extends to a central limit theorem for the random fields Y n (p) = trace n p , p (ζ) = log |ζ − p| with p in the bulk. This is consistent with the hypothesis that Y n comes close to a log-correlated Gaussian field inside the droplet. We refer to [34] for a more thorough discussion of the relationship between Y n , the Gaussian free field, and multiplicative chaos theory in the special case of the Ginibre ensemble. Recently, the fluctuations when f is a characteristic function have been studied in [22] for the Ginibre ensemble. Other relevant references in the context of unitary invariant random matrix ensembles are [15,20,26,35]. We remark also that cases of algebraic potentials have appeared in several works, e.g. [12,13,32,37], while [38,42] investigate the insertion potential V n = |ζ| 2 + 2 Re(tζ) + 2c n log 1 |ζ| , (|t| < 1, c > −1).

1.3.
Outline of the strategy; further results. In the following, we assume that the general conditions on the potentials Q and V n in Subsection 1.1 are satisfied.
Recall that a Hermitian function K is called a positive matrix if the quadratic form N j,k=1 α jᾱk K(z j , z k ) is non-negative for all scalars α i ∈ C and points z i ∈ C. Our starting point is the following structure theorem for limiting kernels. Theorem 1.3. Let K n be the rescaled canonical correlation kernel. There exists a sequence of cocycles c n such that where L n (z, w) is Hermitian-entire, o(1) → 0 locally uniformly on C 2 as n → ∞. The kernels {L n } have the compactness property that each subsequence has a further subsequence converging locally uniformly on C 2 to a Hermitian-entire function L which satisfies the mass-one inequality: Moreover, L is a positive matrix and the inequality L ≤ L 0 holds in the sense of positive matrices, where L 0 is the Bergman kernel of L 2 a (µ 0 ). The kernel L in the theorem is called a limiting holomorphic kernel. We define the corresponding limiting correlation kernel K by K(z, w) = L(z, w)e −V0(z)/2−V0(w)/2 .
We also write R(z) = K(z, z) and speak of a limiting 1-point function or microscopic density.
We have the following "tightness theorem".
Theorem 1.4. Each limiting kernel is nontrivial, in the sense that R does not vanish identically. More precisely, there exists a constant α > 0 such that The kernel K is the correlation kernel of a unique point field which is determined by the collection of p-point functions R p (z 1 , . . . , z p ) = det(K(z i , z j )) p i,j=1 . The point field is the limit of the ensemble {z j } n 1 in the sense of point processes as n → ∞. Here "convergence in the sense of point processes" means convergence of all kpoint functions, cf. [7, Sect. 1.5] and the references therein.
Note the meaning of the asymptotic in (1.16): the first intensity quickly approaches the classical equilibrium density ∆Q as one moves away from the singular point. See Figure 1.
Example. Consider the model Mittag-Leffler ensemble, V n (ζ) = |ζ| 2λ + 2c n log 1 |ζ| . It is interesting to compare the asymptotic in (1.16) with expansion formulas for Mittag-Leffler functions in [24]. To this end, we note that if λ > 1/2 and if |z| 2 > ε > 0 we have by [24, where δ is some number in the interval π/2λ < δ < π/λ and γ(ε, δ) is the contour consisting of the two rays S ±δ,ε = {arg ζ = ±δ, |ζ| ≥ ε} and the circular arc The right hand side in (1.18) has the asymptotic expansion, as z → ∞: [24, eq. (4.7.4)].) By contrast, the asymptotic in [8,Theorem 4] holds for all λ > 0 (and for more general potentials) but gives less precise information when λ > 1/2. If c is negative, the inserted charge is attractive and R = +∞ at the location of the charge, while if c is positive, the inserted charge is repulsive and R = 0 there. (See Figure 1 as well as Section 6).
We note that the special case of Mittag-Leffler ensembles where λ = k is an integer and c = 0 was (except for the name) considered by Chau and Zaboronsky in [14,Section 3]. The limit kernel is there expressed in terms of a Kummer function 1 F 1 rather than the Mittag-Leffler function E 1/k,1/k . The potential Q = |ζ| 2λ also appears in [1] under the name "Freud potential".
To describe a limiting point field, we must determine the corresponding limiting kernel L or, what amounts to the same thing, the correlation kernel K. Our next result, Ward's equation, gives some general information about these kernels. To formulate this result we define (at points where R(z) = 0) the Berezin kernel as well as the Cauchy transform The Berezin kernel B(z, w) expresses the intensity of repulsion from a particle located at z. More precisely, if K is a correlation kernel of a point process Φ and R is its 1-point function, then B(z, w) = R(w) −R z (w), whereR z is the 1-point function for the conditional point process Φ z given that a point of Φ is located at z. See [3, Section 7.6] for more detailed discussion on the Berezin kernel. We now have the ingredients to formulate Ward's equation.
Theorem 1.5. Keep the assumptions from Subsection 1.1 except that the origin is not assumed to necessarily be in the bulk of S. Then each limiting 1-point function R is either trivial (R = 0 identically) or is strictly positive on C * and pointwise on C * and in the sense of distributions on C.
Combining with Theorem 1.4 we see that if 0 is assumed to be in the bulk, then R in Theorem 1.5 is non-trivial, that is, R > 0 at each point of C * and Ward's equation is satisfied.
We stress that C is uniquely determined by R, so Ward's equation gives a feedback relation for the sole unknown function R. Also it should be noted that Ward's equation itself typically has many solutions. To guarantee uniqueness we need to supply suitable apriori conditions on the density R, which depend on the nature of the zooming-point. For example, the same Ward equation holds whether we zoom on a bulk point or a boundary point, and yet the densities look very different.
In this case it is natural to expect that a limiting kernel L be symmetric in the sense that L(z, w) = E(zw) for some entire function E.
From a naive point of view it might seem obvious that each limiting kernel in Theorem 1.6 should be symmetric. The question is however quite subtle; it is connected to the vanishing of a certain entire function G(z), as explained in Section 4. However, if the potential is algebraic (and the droplet is connected), then we are in fact able to show that each limiting kernel is symmetric. Hence Theorem 1.1 will follow as a consequence of Theorem 1.6.
1.4. Yet further results and plan of the paper. In Section 2, we use estimates from the companion paper [8] to prove Theorem 1.3.
In Section 3 we prove Theorem 1.4. We then introduce the distributional Ward equation and prove Theorem 1.5.
In Section 4 we prove Theorem 1.6 and analyze Ward's equation in the radially symmetric case, thus extending the analysis from the papers [5,9].
In Section 5 we complete the analysis of the case when the dominant part Q 0 is radially symmetric by proving apriori symmetry under the hypotheses in Theorem 1.1.
In Section 6 we study the effect of inserting a point charge, by comparing the 1-point functions with and without insertion. It turns out that the difference gives rise to a balayage operation, taking mass from the insertion and distributing it near the boundary, according to a harmonic measure.
In Section 7 we prove Theorem 1.2, modulo an estimate of the 1-point function which is postponed to Section 8.
In Section 8 we prove that, for algebraic underlying potentials, the asymptotics R n (ζ) ∼ nλ 2 |ζ| 2λ−2 as n → ∞ holds to within a very small error, when ζ ∈ Int S is far enough away from the singular point as well as from the boundary. This result can be regarded as a Tian-Catlin-Zelditch type expansion for algebraic insertion potentials, which we were unable to find in the existing literature. We will use the result to complete our proof of Theorem 1.2.

Limiting kernels and their basic properties
In this section we prove Theorem 1.3 on the structure of limiting kernels. We start with Theorem 1.3. The proof will follow easily from the estimates in [8], once the proper notation has been introduced.
Write dµ n = e −nVn dA, (see (1.3)), and let P n = Pol(n) equipped with the norm of L 2 (µ n ). Writing k n for the reproducing kernel of P n , we have (This follows by the classical Dyson determinant formula, which goes back to the early days of random matrix theory, cf. [39].) Now rescale: let r n = n −1/2λ and put z = r −1 n ζ, w = r −1 n η. We define k n (z, w) = r 2+2c n k n (ζ, η).
It is convenient to introduce a "rescaled potential" bỹ V n (z) = nQ(r n z) − 2c log |z| − u(r n z).
Using this notation, the kernel K n in (1.6) can be expressed as Next recall the canonical decomposition (1.5). We recognize that h is the real part of the holomorphic polynomial H whose degree is a positive integer 2d ≤ 2λ.
(Recall that we have assumed that Now consider the factorization where E n and L n are defined by Note that L n is Hermitian-entire while, by Taylor's formula, where V 0 is the microscopic potential in (1.7) and o(1) → 0 as n → ∞, uniformly on compact subsets of C.
Lemma 2.1. Let f n (z, w) be a sequence of Hermitian-entire functions such that f n → 0 locally uniformly on C * 2 . Then f n → 0 locally uniformly on C 2 .
By [8,Corollary 4.7], each subsequence of {L n } has a further subsequence (renamed as L n ) such that L n → L locally uniformly on C * 2 where L is Hermitiananalytic and locally bounded. It is easy to see that L extends to a Hermitian-entire function. Indeed, for fixed w = 0 the function z → L(z, w) has a removable singularity at 0, by Riemann's theorem on removable singularities. Likewise for the functions w → L(z,w) with z = 0. Thus L extends to C 2 \ {0}, and hence to C 2 by the Hartogs' theorem (see [30,Theorem 2.10.1]). In this way, we always regard L as a Hermitian-entire function in the sequel.
The mass-one inequality (1.15) now follows from the following argument. Let us write the canonical decomposition (1.5) as Define a measure µ 0,n by and note that the kernel L n has the following reproducing property, The mass one inequality (1.15) follows from this and Fatou's lemma, since µ 0,n → µ 0 in the vague sense of measures where dµ 0 = e −V0 dA. (By "vague convergence", we mean that µ 0,n (f ) → µ 0 (f ) as n → ∞ for each continuous, compactly supported function f .) We now recognize L n (z, w) as the reproducing kernel for the Hilbert space H n of entire functions defined by Since L n → L and since µ 0,n → µ 0 vaguely, it follows by the arguments in [9, Section 2.3] (or see [6, Section 3.7], [8]) that L is the Bergman kernel of some semi-normed Hilbert space H * of entire functions, which sits contractively inside L 2 a (µ 0 ) (i.e., the natural inclusion is a contraction). Hence L is a positive matrix. Moreover, by Aronszajn's theorem on differences of reproducing kernels in [10,Part I.7], the difference L 0 − L is a positive matrix, i.e., we have 0 ≤ L ≤ L 0 .
We now turn to the kernels E n from (2.1). By Taylor's formula (since we have assumed u(0) = 0) where the second factor in the right hand side is a cocycle. Letting c n (z, w) be the reciprocal cocycle, we thus have the convergence where o(1) → 0 locally uniformly as n → ∞. By this, Theorem 1.3 is proved. q.e.d.

Zero-one law and Ward's equation
Our main goal with this section is to verify Theorem 1.5. We start however with Theorem 1.4.
Proof of Theorem 1.4. The estimate for R(z) as z → ∞ in (1.16) is proved in [8,Theorem 4]. To prove (1.17), we start by noting that the 1-point functions obey a uniform bound of the form R n (z) ≤ M |z| 2c when |z| ≤ 1. Indeed, this follows from Theorem 1.3, via the relation (1.14), on noting that the functions L n (z, z) which converge uniformly to L(z, z) on the unit disc, remain uniformly bounded there.
The statement about convergence of point fields follows from the upper bound 3.1. A distributional Ward identity. We are first going to verify Ward's identity (or loop equation) in the sense of distributions. To this end, it is convenient to use the integration by parts approach from [11,Section 4.2].
As usual, we write Let ψ be a test-function. Interpreting the ∂-derivative in the sense of distributions, we have (for all j) where ∂ j = ∂/∂ζ j and where E n is expectation with respect to the Boltzmann-Gibbs law in (1.4). Summing over j in (3.1) gives We have shown that This is the distributional form of Ward's identity that we need; the point is that ψ is compactly supported in C, not just in C * .

3.2.
Rescaling. (Cf. [6,9].) Fix a test function ψ ∈ C ∞ 0 (C) and observe that Rescaling via z = r −1 n ζ and w = r −1 n η, we define the Berezin kernel and its Cauchy transform by where o(1) → 0 in the sense of distributions on C and uniformly on each compact subset of C * as n → ∞.
Proof. Let ψ n (r n z) = ψ(z). By (3.2) we obtain By changing variables, we calculate each expectation as follows: From the above, we obtain that in the sense of distributions on C. Since This equation holds pointwise on C * and in the sense of distributions on C. Differentiating both sides in the sense of distributions with respect toz, we havē . It remains to note that, in the sense of distributions, where the O-constant is also uniform on each compact subset of C * .

Ward's equation.
We now observe that the Berezin kernel B n can be written as In order to prove Ward's equation, it is convenient to use a formulation in terms of a limiting holomorphic kernel L. We remind of the basic relation for some Hermitian-entireL. Now assume that L(z, z) is a nontrivial kernel which has a zero z 0 which is not isolated, i.e., that there exist distinct zeros z j such that z j → z 0 . Then L(z j , w) = 0 for all w and all j. Noting that L(z, w) is entire in z, we obtain a contradiction.
then g is logarithmically subharmonic some neighbourhood of z 0 .
Proof. Recall first that R does not vanish identically by Theorem 1.4. By Theorem 1.3, we know that L is the Bergman kernel of a contractively embedded, semi-normed Hilbert space H * ⊂ L 2 a (µ 0 ). (This is because of the inequality L ≤ L 0 .) That log L(z, z) is subharmonic now follows from general Bergman space theory, as in [9,Lemma 3.4].
Next assume that L(z, z) = |z − z 0 | 2 g(z) and choose a small neighborhood D of z 0 which does not contain any zero of L(z, z) except z 0 . Since There are now two possibilities: if g(z 0 ) > 0, then we extend log g analytically to z 0 and have ∆ log g(z 0 ) ≥ 0; if g(z 0 ) = 0, then log g(z 0 ) = −∞ and log g(z) again satisfies the sub-mean value property in D.
We now set out to find suitable subsequential limits of the Berezin kernels B n , defined in (3.3). For this, we fix a subsequence L n which converges locally uniformly to a limiting holomorphic kernel L.
The main observation is that if L(z 0 , z 0 ) > 0, then the convergence is uniform for all (z, w) ∈ D × K where D is some neighbourhood of z 0 and K is a given compact subset of C. To see this, it suffices to note that , and that L(z, z) = lim L n (z, z) ≥ const. > 0 in a neighbourhood of z 0 .
We need to check that the convergence B n → B implies a suitable convergence C n → C on the level of Cauchy transforms. For this purpose, we formulate the next lemma.
Lemma 3.4. Suppose that R(z) = L(z, z)e −V0(z) does not vanish identically. If Z is the set of isolated zeros of L(z, z), then C n → C locally uniformly on C\(Z ∪{0}) as → ∞. Moreover, the function z → zC(z) is bounded on V \Z for each compact subset V ⊂ C.
Proof. Fix a small number > 0. We define a compact subset K of C 2 by The remarks preceding the lemma show that we can find N such that if ≥ N then Let us recall also that there is a constant M such that, if z ∈ Z , then For ≥ N and z with dist(z, Z ∪ {0}) ≥ and |z| ≤ 1/ , we now obtain Here, we have applied the mass-one inequality (1.15) to estimate the integral over {|z − w| > 1 } and the inequality (3.4) has been used for the estimate of the integral over {|z − w| < 1/ }. To show the local boundedness of the Cauchy transform C(z), we fix a compact subset V of C and a number = V such that V ⊂ {|z| < 1/ }. Let us write c = − min{c, 0}.
Let δ be an arbitrary small number. Using the estimate (3.5), we see that for This shows that |C n (z)| ≤ M |z| −1 for all z ∈ V \ (Z ∪ {0}), where the constant M depends only on V .
Finally, the following lemma concludes the proof of Theorem 1.5. Proof. By Lemma 3.1, we have where o(1) → 0 as a distribution on C and uniformly on each compact subset in C * . For a compact subset V of C, we moreover know that V ∩ Z is a finite set and zC n (z) → zC(z) boundedly and locally uniformly on V \ (Z ∪ {0}). For a test function φ and a small number , we take V , V such that Here, V is compact and the (normalized) area of V \ V is less than N for some N > 0. Moreover, since |C n (z)| ≤ M |z| −1 for all z near 0 as in the proof of Lemma 3.4, we get Thus the integral can be made small since C n → C uniformly on V and zC n (z) is uniformly bounded on V \ (Z ∪ {0}). Thus C n → C in the sense of distributions on C, which implies∂C n →∂C in the same sense. In view of (3.7) and the convergence R n → R such that R n (z) ≤ M |z| 2c for |z| ≤ 1 and R n (z) are uniformly bounded in each compact subset of C * , we conclude that the measures ∆ log R n converge to ∆ log R. Passing to the limit as → ∞ we obtain (3.6) in the sense of distributions. We now prove that R > 0 on C * . For this, we suppose R(z 0 ) = 0 for some z 0 = 0. Then by Lemma 3.2, for some Hermitian-entire function L 1 . We fix a small disk D ⊂ C * centered at z 0 where L 1 (z, z) is logarithmically subharmonic (cf. Lemma 3.3), and define two measures ν and ν 1 by By Lemma 3.3, these measures are both positive. Now note that ν = ν 1 + δ z0 and consider the Cauchy transform Since∂C ν = ∆Q 0 + ∆ log R on D, the Ward's equation (3.6) gives∂ for some smooth function v on D. Since C is bounded in D \ {z 0 } by Lemma 3.4, C ν (z) remains bounded as z → z 0 . This implies that ν({z 0 }) = 0, which contradicts that ν 1 is positive. The contradiction shows that R(z 0 ) > 0.
The positivity of R implies that log R is real-analytic on C * , so the right hand side of (3.6) is smooth there. By Weyl's lemma which asserts that every weak solution of Laplace's equation is smooth, C(z) is also smooth on C * and hence Ward's equation holds pointwise on C * .

Symmetric solutions in the dominant radial case
We will now prove our first principal result, Theorem 1.6. Our proof elaborates on arguments from the papers [5,9]. We will initially allow for kernels L which are not necessarily rotationally symmetric.
We start by noting that if R is a non-trivial limiting 1-point function in Theorem 1.5, then Ward's equation can be written in the form where L is the corresponding limiting 1-point function, R(z) = L(z, z)e −V0(z) .
We here consider the dominant radial case and we write dµ 0 = e −V0 dA.
Since the kernel L is Hermitian-entire, we can represent it in the form a jk z jwk , (a jk =ā kj ).
We know also that L is nontrivial, i.e., L(z, z) > 0 for all z = 0 by Theorem 1.5. Let L 0 be the Bergman kernel for the space L 2 a (µ 0 ). Since Q 0 is radially symmetric, we have the formula .
We want to prove that L = L 0 .
Let us rewrite Ward's equation. Below we fix a complex number z = 0.
The Cauchy transform C(z) is computed as follows: From the fact that we obtain that (all sums are over all j, k, l, m unless otherwise is specified) This can be rewritten It follows that∂ Hence Ward's equation, which can be written in the form is equivalent to just∂ (S 2 (z) − ∂ log L(z, z)) = 0.  The Taylor coefficients g j in G(z) = g j z j depend on a jk in a complicated way. Comparing coefficients ofz l in (4.3), we deduce for each l ∈ Z + the identity Thus, for example, the constant term g 0 = G(0) obeys infinitely many relations a ml a 0,m−1 z m−1 2 L 2 (µ0) , l = 0, 1, 2, . . . .
Due to the formidable appearance of these relations, we now abandon the quest for the most general limiting kernel L and restrict our attention to symmetric ones.
We thus assume that L(z, w) = a jk z jwk is rotationally symmetric, i.e., we assume that a jk = a j δ jk for some numbers a j . In this case, the system (4.4) becomes: for all m ≥ 1 By considering the smallest m ≥ 1 such that a m = 0 we conclude that G ≡ 0, i.e., Before proving Theorem 5, recall that L satisfies the mass-one inequality (1.15). The mass-one inequality can be written as  Proof. We use (4.5), (4.6), and induction on m ≥ 0. Put N m = inf{j ; j ≥ m, a j = 0} and note that, since L(z, z) ∼ ∆Q 0 (z)e V0(z) as z → ∞ (by Theorem 1.4), we have N m < ∞ for each m.
A comparison of the formula (4.1) for the kernel L 0 with the above lemma shows that L = L 0 , finishing our proof of Theorem 1.6. q.e.d.

Universality at radial type singularities
We now prove Theorem 1.1. Suppose that the potential V n is of the form where Q r is radially symmetric and Q r (z) = |z| 2λ + O(|z| 2λ+ ) as z → 0 for some > 0 and where h is a harmonic polynomial with h(0) = 0. We assume that the degree of h is at most d, where, without loss of generality, d ≥ 2λ.
We suppose in addition that 0 ∈ Int S, that S is connected, and that ∂ Pc S is an everywhere regular Jordan curve. Now write h = Re H where H is the holomorphic polynomial The crucial property that we shall need from Q r is that ζ∂Q r (ζ) ∈ R for all ζ. This is satisfied since Q r is radially symmetric.
Let p n,j be the j:th orthonormal polynomial with respect to e −nVn and rescale about 0 by letting q n,j (z) = r 1+c n p n,j (r n z),Ṽ n (z) = nQ(r n z) − 2c log |z|. WithH n (z) = nH(r n z) we can then define a "rescaled holomorphic kernel" by L n (z, w) = n−1 j=0 q n,j (z)q n,j (w)e −(Hn(z)+Hn(w))/2 .
By a straightforward extension of normal families argument (see Section 2) we obtain easily that each subsequence of L n has a further subsequence converging locally uniformly to a limiting holomorphic kernel L. (The terms inH n of degree > 2λ are negligible on compact sets. ) We want to prove the asymptotic rotation invariance ∂ θ L n (z, z) → 0 as n → ∞ along any subsequence. To prove this we observe that Writeh n = ReH n . Then (z∂q n,j (z) − zq n,j (z)∂h n (z))q n,j (z).
Since ∂h n is a holomorphic polynomial of degree at most d − 1, zq n,j ∂h n is a polynomial of degree at most j+d. Hence it can be expressed as a linear combination of the polynomials q n,l : zq n,j ∂h n (z) = j+d l=0 zq n,j ∂h n , q n,l q n,l (z) where f, g = fḡe −Ṽn dA. Now setQ r = nQ r (r n z) = |z| 2λ + · · · . An integration by parts gives zq n,j ∂h n , q n,l = − zq n,j (z)q n,l (z)∂(e −hn(z) )e −Qr+2c log |z| dA(z) = q n,j , q n,l + z∂q n,j , q n,l + zq n,j (z)q n,l (z)(−nr n ∂Q r (r n z) + c z )e −Ṽn(z) dA(z).
We then have the identity zq n,j ∂h n , q n,l = (1 + c) q n,j , q n,l + z∂q n,j , q n,l − α j,l and from (5.1), α j,l q n,l (z)q n,j (z)).
In order to estimate the sum in the right hand side of (5.3), we need a good control of the polynomials q n,j with |j − n| ≤ d. For this purpose, we will employ recent results on asymptotics for the orthonormal polynomials p n,j with respect to e −nVn , to obtain the following result.
Lemma 5.1. Let K be any fixed compact subset of Int Pc S. Then for any given integer κ ≥ 0 and any j = j(n) with |j − n| ≤ d we have Proof. We will use one of the main results from the paper [25] 1 , which implies that if V n is any potential of the type indicated above, then for any fixed κ ≥ 0 we have Here χ is a smooth function which equals to 1 in a small tubular neighbourhood of the outer boundary of S while χ = 0 in the complement of a slightly larger tubular neighbourhood, which is still small enough that χ = 0 near all the logarithmic singularities occurring in the potential.
Following [25] we write Q for (the analytic continuation of) the bounded holomorphic function on C \ Pc S satisfying Re Q = Q on ∂ Pc S. (We fix Q uniquely by requiring that the imaginary part vanishes at infinity.) The function F n,j is of the form where φ is the analytic continuation of a univalent mapĈ \ Pc S →Ĉ \ D and B p are certain holomorphic functions, bounded on the support of χ. It is clear that the asymptotic in (5.4) implies the statement in the lemma.
Remark. In [25] there is an assumption that the potential Q be everywhere C 2smooth in the plane. It is however easy to see that the results hold under the assumptions in the present paper. Indeed, the details of the potential in a given compact subset of Int S are completely irrelevant in [25] (as long as it is admissible in the sense of [40]). Similarly, the details of the potential in the exterior, away from the boundary, are not relevant. The only point where any kind of higher regularity is used is in the form of real-analyticity near the boundary.
Proof. By a change of variables we have (5.5) α j,l = n z∂Q r (z)p n,j (z)p n,l (z) e −nVn(z) dA(z).
Since the |p n,j | 2 e −nVn are negligible outside the support of χ (see remark below) we can restrict to some small neighbourhood of the droplet. We can then bound the integral in (5.5) by C |p n,j ||p n,l |e −nVn , so by the Cauchy-Schwarz inequality it follows that |α j,l | ≤ Cn.
Remark. To make the proof above precise, we here provide an exterior estimate of |p n,j | 2 e −nVn on C \ (supp χ ∪ Pc S), i.e., outside of a neighborhood of Pc S. Indeed, a modification of the pointwise asymptotics in Theorem 1.3.5 from [25] gives the asymptotics which is uniform for z outside a neighborhood of S. Moreover, according to the terminology in the paper [25], Re Q + log |φ| 2 =Q, which gives a very fast decay of |p n,j | 2 e −nVn outside an arbitrarily small neighborhood of Pc S. Now let K be a given compact subset of C * . We shall estimate the integral Hence by Lemma 5.1, we have J n = O(n −κ ) for any given κ > 0. (The compact set K corresponds under the rescaling to the dilated set r n K which is surely bounded away from the outer boundary when n is large enough.) It follows that if L = lim L n l is a limiting holomorphic kernel, then for each compact set K ⊂ C, K |∂ θ L(z, z)| dA(z) = 0. This is only possible if ∂ θ L(z, z) = 0 identically, i.e., L is rotationally invariant.
The proof of Theorem 1.1 is complete. q.e.d.

Insertion as a balayage operation
In this section, we will discuss the effect of insertion of a point charge, by comparing the first intensity of the process {ζ j } n 1 with respect to the inserted point charge c to the intensity of the corresponding process with c = 0. Our discussion elaborates on the concluding remarks from [3, Section 7.7].
We will now exploit the freedom of choosing the perturbation u in the potential V n = Q − 2c n − 1 n u, where we write (ζ) = log |ζ|. Here, we take u to be smooth and harmonic. One natural choice is to put u = 0, i.e., to set We will call (6.1) the "pure log-normalization" of V n . Another interesting choice, the "Green's form", is to set In order to apply the theory of [3,4] we need to impose a few conditions on the geometry and topology of the droplet S. First of all, we recall our assumption that Q be real-analytic in a neighbourhood of S with exception at the origin. (See (1.2).) We will now also impose the condition that ∆Q is strictly positive in a neighbourhood of the boundary ∂S. It is well-known that this guarantees that S has finitely many components, and that ∂S is a finite union of real-analytic curves, possibly having finitely many singular points of known types (cusps or double points). (Cf. [7,37] and references.) We shall assume that S is connected, and that the boundary is everywhere smooth. In this case, the Green's function G(ζ) can be extended to a smooth function on C * with G ≡ 0 near ∞. We fix such an extension and insist on calling it G.
Let R n andR n be the 1-point functions associated, respectively, with the external potentials Q and V n . We shall measure the effect of the insertion by studying the difference ρ n =R n − R n , where we use the convention that R n is identified with the measure R n dA (and likewise forR n ).
Note that the asymptotics of ρ n (f ) can be nontrivial only if the support of f contains either the point 0, or some portion of the boundary of S. Indeed, if we exclude neighbourhoods of 0 and of the boundary, then on the rest of the plane both R n andR n are very close to n∆Q · 1 S , in the sense that the difference is negligible, cf. [6,8].
In order to simplify the following discussion, we assume now that the potential Q has a dominant radial bulk singularity at 0 of Mittag-Leffler type, and that the coefficient τ 0 = 1. That is, we assume according to the canonical decomposition (1.5) that Q 0 = |z| 2λ , V 0 = |z| 2λ − 2c log |z|. In addition we will assume that each limiting kernel is rotationally symmetric, that is (by Theorem 1.6) each limiting kernel is equal to L(z, w) = λ · E 1/λ,(1+c)/λ (zw). Finally, we assume that there are no other singular points in the bulk, except the one at 0.
In the rest of this section, we assume that all of the above conditions are satisfied. For a domain Ω bounded by a Jordan curve γ in C, the harmonic measure with respect to Ω evaluated at a point z ∈ Ω is the hitting probability distribution on γ of a Brownian motion starting at z. We refer to [23] for other equivalent definitions of the harmonic measure.
Here δ 0 is the Dirac measure at 0 and ω 0 the harmonic measure with respect to S, evaluated at 0.
where ω ∞ is the harmonic measure with respect to C \ S, evaluated at ∞.
The theorem shows that the insertion of a point mass, corresponding to different natural boundary conditions, gives rise to different kinds of balayage operations, see Figure 2, cf. [23,40] for the basic facts about balayages. In other words, ρ n → −cδ 0 in the interior of the droplet.
Proof. Rescaling about 0 on the scale n −1/2λ , we obtain the 1-point intensities R n andR n which, by Theorem 1.6 have known locally uniform limits R andR. We shall show that the total mass of the measure R −R is In view of well-known bulk estimates of R n andR n away from a microscopic neighbourhood of 0, this will prove the lemma. Before proceeding with the proof, we pause to recall some details about the estimates that come into play here.

Consider the annular regions
where M is large and d is some small constant so that B n lies inside the droplet. We claim that the total variation of ρ n on A n ∪ B n can be made less than any given > 0 by choosing large M and small d. Indeed, for ζ ∈ A n , the asymptotics from [8, Section 3] shows that there exist some constants c, C such that for all large n. Thus where the right hand side can be made small by taking M large enough. By a slight adaptation of the Hörmander estimate in [6, Section 5], we obtain the following estimate. Lemma 6.3. There exist some constants c, C > 0 such that for ζ ∈ B n (6.5) for all large n.
The estimates in (6.5) show that for ζ ∈ B n |R n (ζ) − R n (ζ)| ≤ C 1 + n max(1, 1 λ ) e −c(log n) 2 is bounded, and the total variation of ρ n over B n can be made as small as we want by choosing a small d. On the other hand, away from a small neighborhood of 0, the asymptotic formula (cf. [3,4]) where O(n −1 ) is uniform on each compact subset in Int S \ {0} can be used. We have thus reduced our problem to proving the identity in (6.4).
To show (6.4) we recall by Theorem 1.4 that both R(z) andR(z) approach ∆Q 0 (z) quickly as z → ∞. In particular, the integrals are convergent; if we can compute them, we can obtain (6.4) by a simple subtraction.
Our problem is thus to evaluate the integral which is the same as We first consider the case when λ is a rational number. The computation of the integral can be reduced to the case λ = 1 by the following observation.
A rational number λ with λ > 0 is expressed as the fraction p/q of two integers p, q ≥ 1. Letting j = mp + l for some integers m, l with 0 ≤ l ≤ p − 1, we have dt.
Writing F s (t) = e −t t c+s E q,c+1+s (t q ), 0 ≤ s ≤ q − 1, we have e −t t c E 1,c+1 (t) = F 0 (t) + · · · + F q−1 (t) and F s = F s−1 − F s for all 1 ≤ s ≤ q − 1. This implies that Using the asymptotic formula (cf. [24, eq. (4.4.18)]) we now obtain F s (∞) = 1 q for 0 ≤ s ≤ q − 1. In view of equation (6.8), we thus obtain for c > −1 If we temporarily accept that I c,1 = c for c > −1, we now deduce that and hence The identity (6.7) now follows by the continuity in λ for all real numbers λ > 0, provided that we can show it for λ = 1.
It remains to verify that I c,1 = c, where Our verification that I c,1 = c is somewhat lengthy, and works in fact for complex c with Re c > −1. An alternative short proof in the case when c is a positive integer is given in the remark below. We first assume that Re c > 0. To show that I c,1 = c, it is convenient to call on the lower and upper incomplete gamma functions We claim that (for Re c > 0 and t > 0) To see this, we integrate the right hand side by parts, To repeat the integration, we similarly observe that for all ν = 0, 1, 2, . . . s c+ν e −s ds, and note that, for fixed t > 0, (6.12) γ(c + ν, t) Γ(c + ν + 1) → 0, (ν → ∞).
In view of (6.10)-(6.12) we have and (6.9) is proved. It follows from (6.9) that (6.13) To calculate this, we note that (6.14) Combining (6.13)-(6.15) we find that For the case when −1 < Re c < 0, we write Thus, we have finishing the proof of the lemma.
Remark. When c is a positive integer, we can easily prove the identity I c,1 = c as follows. We have It follows that In order to see the effect of the insertion near the boundary, we will apply the boundary fluctuation theorem in [4].
For a suitable function h, let V n = Q − 2h/n be the perturbed potential and R h n be the corresponding 1-point function. Write ρ h n for the difference R h n − R n . We define for any suitable function f , a new function f S in the following way: f S = f on S and f S equals the harmonic extension of f | ∂S toĈ \ S on that set. Theorem 6.4. Suppose that f is a smooth test-function with compact support, which vanishes in a small, fixed neighborhood of 0 and h is C 2 -smooth outside the origin and has compact support. Then there exists a small number α > 0 such that Proof. The estimate (6.16) is given in [4,Theorem 2.3], except for the order of the error-term O(n −α ). To obtain this error-term, we just need to carefully examine the proof in [4]. In the following lines we will explain how this can be seen, temporarily borrowing notation from [4]. (The reader who wishes to understand the following details should thus have a copy of [4] at hand.) Let v be a suitable test-function, smooth in a neighbourhood of the boundary ∂S, satisfying conditions (3.4-i)-(3.4-iii) from [4].
The main tool behind the convergence of fluctuations in [4] is the limit form of Ward's identity: where ε 1 n [v] and ε 2 n [v] are certain error-terms given in [4]. By Prop. 4.5 and Prop. 4.6 in [4], we have for some small β > 0, and it is easy to show that the term (σ h n − σ)(∂v/2 + 2v∂h), is O(δ n ) where δ n = log 2 n/ √ n. Hence it follows that for some α > 0. Now we assume that f is a suitable function supported near the boundary. Write as in [4,Section 5]. Then (6.17) implies that, Hence the fluctuation term ν h n (f ) obeys the asymptotic and we obtain The general case f = f 0 + f + + f − is immediate from this, as in [4,Section 5].
To prove Theorem 6.1, we consider the cases when h = G and when h = . The function G is an extension of the Green's function, which is smooth on C * and has compact support. For the case of "pure log-normalization", the limit form of Ward's identity in [4] can be applied and Theorem 6.4 holds for since the function does not grow too fast and ∂ (z) ∼ z −1 near ∞.
Proof of Theorem 6.1. Assume that the external potential V n has the Green's form in (6.2). We then have ρ n = ρ G n .
Lemma 6.5. Suppose that f is a smooth test-function which vanishes in some small, fixed neighbourhood of 0, and let ω 0 be harmonic measure for S evaluated at 0. Then Proof. The assumptions on f imply that we can apply the boundary fluctuation theorem 6.4. Also let G be the Green's function in (6.3), so G S = 0 in C \ S.
The asymptotic formula for the variance of fluctuations in Theorem 6.4 implies that, for the test-functions f under consideration, since f vanishes near 0. Here ∂/∂n is differentiation in the outwards normal direction. We refer to the argument in [3, p. 76] for details about the calculation. The lemma follows, since − 1 2π ∂G ∂n ds is the harmonic measure ω 0 (see [23]).
Combining lemmas 6.5 and 6.2, we conclude the proof of part (i) of Theorem 6.1. Now assume that the external potential V n has the pure log-form, i.e., In this case, we have ρ n = ρ n . Let G ∞ be the Green's function of C \ S with pole at ∞ (so G ∞ (ζ) ∼ log |ζ| as ζ → ∞). We consider G ∞ as being extended to C in some smooth way. Note that and that S is harmonic onĈ \ S. We now fix a function f ∈ C ∞ 0 which vanishes near 0 and apply the result of [4] as in the proof of Lemma 6.5. The result is this time that where ω ∞ is harmonic measure of C \ S evaluated at ∞. Combining with Lemma 6.2, we conclude the proof of part (ii) of Theorem 6.1.

A Central Limit Theorem
In this section we prove Theorem 1.2. Our strategy is to first give the proof in the simplified model Mittag-Leffler case, and after that we will adapt the proof to a general algebraic insertion potential. Lemma 7.1. Suppose that V n is the model Mittag-Leffler potential Then the random variables X n = 2 √ log n (trace n − E n trace n ) converge in distribution to the centered normal distribution with variance 1/λ.
We remind the reader of the notation (ζ) = log |ζ|, trace n = (ζ j ) where {ζ j } n 1 is a random sample. E n denotes the expectation with respect to the potential V n , see (1.4). We also recall that the rescaled one-point function of (7.1) equals to (See the example in Section 1.) Proof of the lemma. We shall apply the variational approach from [4] with the scale of potentials Here t is a fixed real constant and n is large enough so that c n,t > −1.
The method uses the function where E n is expectation with respect to the potential V n . As in [3,4] we note that where E n,t is expectation with respect to potential V n,t . Hence where R n,t and R n are 1-point functions with respect to V n,t and V n , respectively. Inserting explicit expressions (see (7.2)) we obtain )e −s ds.
Let ψ be the polygamma function, The computations above show that We now use Taylor's formula to write where ξ j is some number between (j + c)/λ and (j + c + c n,t )/λ, and have To proceed with this, we note the following lemma.
Lemma 7.2. ψ and ψ have the series expansions The proof of the lemma is an immediate consequence of Weierstrass' form of the Gamma function, Since ψ is decreasing for x > 0, we obtain Lemma 7.2 shows that as x → ∞. Then it follows from (7.5) and (7.6) that We also have t 2 2(λ 2 log n) 3/2 n j=1 ψ (ξ j ) → 0 as n → ∞ since ψ (x) = −1/x 2 +O(x −3 ) for large x. By (7.4) we now obtain that F n (t) → t/λ as n → ∞, and it is easy to see that our estimates give locally uniform convergence on R. Since F n (0) = 0, we conclude that F n (t) = t 2 /2λ as n → ∞, i.e., E n e tXn → e t 2 /2λ .
It is well-known that this implies convergence in distribution to the normal distribution with mean zero and variance 1/λ.
We now generalize to an arbitrary potential of the form where h is a harmonic polynomial (in some neighbourhood of the droplet). Let R n and R n denote the 1-point function and the rescaled one with respect to V n . We know that the rescaled 1-point function R n converges to However, in order to apply the argument in Lemma 7.1, it will be better to work with the 1-point function with respect to V n (ζ) = |ζ| 2λ − 2c n log |ζ|, which is the model Mittag-Leffler case considered in Lemma 7.1. Note also that Theorem 1.1 implies the convergence R n − R n → 0 in the sense of distributions on C and locally uniformly on C * . Now fix a suitable t ∈ R and put c n,t = t √ log n .
We consider the perturbed potential V n,t = V n − 2c n,t n log |ζ| and we write E n,t , R n,t for the corresponding expectation and 1-point function, respectively. We also write R n,t , R n,t for the corresponding rescaled 1-point functions, i.e., R n,t and R n,t are the rescaled 1-point functions associated with the potentials V n,t and V n,t , respectively. As before, we introduce where X n = 2 √ log n (trace n − E n trace n ), ( (ζ) = log |ζ|).
Consider the set R n = {ζ ∈ Int S; |ζ| > r n (log n) 1 λ , dist(ζ, ∂S) > r n (log n) 1 λ }. We shall use the uniform estimate (7.9) sup ζ∈Rn |R n (ζ) − n∆Q 0 (ζ)| ≤ Ce −α log 2 n with some constants C, α > 0. The detailed proof of (7.9) depends on an adaptation of the technique of approximate Bergman projections, which is postponed to the next section (see Theorem 8.1 below). By the same token, the obvious counterpart to (7.9) is true also for the difference R n,t − n∆Q 0 , so we obtain the result that In the vicinity of the boundary ∂S, we do not have such a strong uniform control, but due to our discussion of balayages in Section 6 we know that R n,t −R n ∼ c n,t ω ∞ there, where ω ∞ is harmonic measure of ∂S evaluated at ∞.
Outside a neighborhood of S, the one point function is controlled by the exterior estimate, see e.g., [4, Section 4.1.1]. For any m > 0, there exists a constant C m such that R n,t (ζ) ≤ C m n −m (1 + |ζ|) −3 for all ζ outside a fixed neighborhood of S.
Combining these asymptotic estimates, we find that (7.11) |ζ|>rn(log n) Here the O-constant is proportional to |t|. The convergence R n,t −R n → c n,t (ω ∞ − δ 0 ) shown in Theorem 6.1 gives (7.12) 2 log r n √ log n |ζ|>rn(log n) Here, the error bound o(1) is obtained by the error bound of the convergence in Theorem 6.4. Let us denote By (7.11) we have that Let us introduce the function n (z) := 2 log |z| √ log n χ D(0;(log n) 1/λ ) (z). Changing variables in (7.8) by ζ = r n z and the asymptotic expansion (7.12) give We split the last expression as We prove by Lemma 7.1 that C n,t = o(1) as n → ∞.
Indeed, by the approximations above applied to suitable model Mittag-Leffler ensembles, we obtain 2 √ log n C log |r n z| · (R n,t (z) − R n (z)) = C n,t + t λ + o (1) and the integral converges to t/λ by Lemma 7.1. We now want to show that A n,t , B n → 0 as n → ∞. For this, we recall the following estimates, which are proved in [8].
First of all, note that R n − R n → 0 and R n,t − R n,t → 0 locally uniformly on C * and in the sense of distribution on C. (See Theorem 1.6 and Theorem 1.1.) Also note that Theorem 1.3 implies that these 1-point functions are dominated near the origin (R n (z) ≤ C|z| 2c , |z| ≤ 1). Thus, there is no problem to estimate the integral of n · (R n − R n ) over a large compact disk |z| ≤ M .

Asymptotics for the 1-point function
In this section, we will be dealing with the class of generalized Hele-Shaw potentials, by which we mean potentials of the form (8.1) V n (ζ) = |ζ| 2λ − 2c n log |ζ| + 2 Re H(ζ) where c > −1 and where H is holomorphic in a neighbourhood of the droplet. We assume, as always, that 0 is an interior point of the droplet.
Note that Theorem 8.1 in particular completes our argument for the CLT in Section 7 (i.e., we obtain the missing estimate (7.9)).
Before discussing the proof, it is instructive to compare the result with other kinds of approximations.
To prepare the ground, we fix a sequence ζ n with ζ n ∈ R n , and we put δ n := c λ n −1/2 |ζ n | 1−λ log n for a small enough constant c λ less than 1/ max{λ, 10}. If λ = 1, the distance δ n = O(n −1/2 log n) is chosen so as to be "sufficiently large" compared with the typical inter-particle distance n −1/2 . In general, our choice of δ n is used to deduce the inequality (8.3), which will be seen to imply the exponential decay in Theorem 8.1. We can easily see that δ n /|ζ n | = c λ n −1/2 |ζ n | −λ log n ≤ c λ , ζ n ∈ R n .
We also fix a sequence of cut-off functions χ n with χ n = 1 on D(ζ n ; 2δ n ), χ n = 0 outside D(ζ n ; 3δ n ) and ∂ χ n L 2 ≤ C (independent of n).
The following submean-value property where C and C are some constants depending on t.