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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.

For a function ϕ in L 2 (0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates ϕ(nx), n = 1, 2, 3, . . . , constitutes a Riesz basis or a complete sequence in L 2 (0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f (s) = n a n n −s , where the coefficients a n are square summable. It proves useful to model H as the H 2 space of the infinite-dimensional polydisk, or, which is the same, the H 2 space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given f in H and characters χ, f χ (s) = n a n χ(n)n −s is a vertical limit function of f . We study certain probabilistic properties of these vertical limit functions.

We consider weight functions Q : ℂ→ℝ that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one‐component plasma (2‐dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m→t, where 0

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.