Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

TL;DR

This paper studies the universal behavior of a two-dimensional charged particle system near the boundary of a smooth domain, revealing how the kernel's limit depends on the ratio of particles to charge and connecting it to known universal kernels.

Contribution

It establishes the universality of boundary kernels for charged particle ensembles with potential theoretic weights, depending on the ratio N/s, and relates these to conformal maps and known kernels.

Findings

The limiting kernel depends on the ratio N/s.

Universality holds for kernels near smooth boundary points.

Special case N/s → 0 recovers Lubinsky's kernel.

Abstract

We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to \beta = 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N \to \infty, of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N/s, and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point \zeta \in \partial K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement K to the complement of the closed unit disk. When N / s \to 0 we recover the universal kernel discovered by Doron Lubinsky in Universality type limits for Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154, 2010.