In this note, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane -the "droplet". We give two proofs for the Gaussian field convergence of fluctuations of linear statistics of eigenvalues of random normal matrices in the interior of the droplet. We also discuss various ramifications of this result.2000 Mathematics Subject Classification. 15B52.

We study existence and universality of scaling limits for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's equation -an integro-differential identity satisfied by the rescaled one-point function.

Abstract. Consider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of fluctuations, and we prove that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.Given a suitable real "weight function" in the plane, it is well-known how to associate a corresponding (weighted) random normal matrix ensemble (in short: RNM-ensemble). Under reasonable conditions on the weight function, the eigenvalues of matrices picked randomly from the ensemble will condensate on a certain compact subset S of the complex plane, as the order of the matrices tends to infinity. The set S is known as the droplet corresponding to the ensemble. It is well-known that the droplet can be described using weighted logarithmic potential theory and, in its turn, the droplet determines the classical equilibrium distribution of the eigenvalues (Frostman's equilibrium measure).In this paper we prove a formula for the expectation of fluctuations about the equilibrium distribution, for linear statistics of the eigenvalues of random normal matrices. We also prove the convergence of the potential fields corresponding to corrected fluctuations to a Gaussian free field on S with free boundary conditions.Our approach uses Ward identities, that is, identities satisfied by the joint intensities of the point-process of eigenvalues, which follow from the reparametrization invariance of the partition function of the ensemble. Ward identities are well known in field theories. Analogous results in random Hermitian matrix theory are known due to Johansson [13], in the case of a polynomial weight. By D(a, r) we mean the open Euclidean disk with center a and radius r. By "dist" we mean the Euclidean distance in the plane. If A n and B n are expressions depending on a positive integer n, we write A n B n to indicate that A n ≤ CB n for all n large enough where C is independent of n. The notation A n ≍ B n means that A n B n and B n A n . When µ is a measure and f a µ-measurable function, we write µ( f ) = f dµ. We write ∂ = 1 2 (∂/∂x − i∂/∂y) and∂ = 1 2 (∂/∂x + i∂/∂y) for the complex derivatives. General notation.

Abstract. We consider fairly general weight functions Q : C → R, and let K m,n denote the reproducing kernel for the space H m,n of analytic polynomials u of degree at most n − 1 of finite norm u 2 mQ = C |u(z)| 2 e −mQ(z) dA(z), dA denoting suitably normalized area measure in C. For a continuous bounded function f on C, we consider its (polynomial) Berezin transformFor a parameter τ > 0 we prove that there exists a compact subset S τ of C such thatfor all continuous bounded f if z is in the interior of S τ ∩ X. Equivalently, the measures B z m,n converge to the Dirac measure at z. The set S τ is the coincidence set for an associated obstacle problem.We also prove that the convergence in (0.1) is Gaussian when z is in the interior of S τ ∩ X, in the sense that withdP(ζ) = e −|ζ| 2 dA(ζ) denoting the standard Gaussian. In the "model case" Q(z) = |z| 2 , S τ is the closed disk with centre 0 and radius √ τ. We prove that if z is fixed with |z| > √ τ, the corresponding measures B z m,n converge to harmonic measure for z relative to the domain C * \ S τ , C * denoting the extended plane.Our auxiliary results include L 2 estimates for the ∂-equation ∂u = f when f is a suitable test function and the solution u is restricted by a growth constraint near ∞.Our results have applications e.g. to the study of weighted distributions of eigenvalues of random normal matrices. In the companion paper [1] we consider such applications, e.g. a proof of Gaussian field convergence for fluctuations of linear statistics of eigenvalues of random normal matrices from the ensemble associated with Q.

We give a method for taking microscopic limits of normal matrix ensembles. We apply this method to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.2010 Mathematics Subject Classification. Primary: 60B20. Secondary: 60G55; 81T40; 30C40; 30D15; 35R09.