Scaling limits of random normal matrix processes at singular boundary points

Abstract

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.

Rescaling Ward Identities in the Random Normal Matrix Model

“…Our first theorem states the existence of sequential limits of the rescaled point processes Θ n and specifies the form of limiting correlation kernels. With a view to later applications [5] we shall adopt a somewhat broader point of view, rescaling about a moving point p = (p n ) ∞ 1 where p n is a sequence of points in S. A constant sequence p n = p will be identified with the point p. In the following result we are also at liberty to choose a sequence of angles θ n ∈ R and rescale about p according to (1.6) z j = e −iθn n∆Q(p n ) (ζ j − p n ) .…”

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

“…This model was introduced by Balogh and Merzi [13] who showed that the equilibrium measure is supported on the interior of the domain, S = {λ : |λ d − t| ≤ t c } where t c = d −1/2 . Due to the lemniscate type shape of S, the random point configuration corresponding to (1.4) has been referred to as the lemniscate ensemble [7], see Figure 1. When t passes through t c , the topology of the droplet undergoes a transition from being simply connected to consisting of d connected components.…”

“…When t passes through t c , the topology of the droplet undergoes a transition from being simply connected to consisting of d connected components. The case t = t c has been singled out as a model with a non-trivial boundary singularity (see [7]) and the effect of this singularity on the partition function asymptotics is of interest (see also Remark 3.9).…”

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

“…In a very recent work [27] the asymptotic of orthogonal polynomials for a varying planar measure was obtained in the regular case. Critical regimes have been considered in [34], in [9,32] studying a normal matrix model with cut-off, and in [1] where new types of determinantal point fields, have emerged. The equilibrium problem for a class of potentials to which our case belongs, was also recently considered in [2].…”