The random normal matrix model: insertion of a point charge

TL;DR

This paper investigates the microscopic behavior of eigenvalues in a 2D normal matrix model near a point charge, revealing universal scaling limits and applications to characteristic polynomial distributions.

Contribution

It characterizes all rotationally symmetric scaling limits ('Mittag-Leffler fields') and proves their universality for algebraic potentials in the model.

Findings

Identification of all rotationally symmetric scaling limits.

Proof of universality for Mittag-Leffler fields with algebraic potentials.

Asymptotic distribution results for the logarithm of characteristic polynomials.

Abstract

In this article, we study microscopic properties of a two-dimensional eigenvalue ensemble near a conical singularity arising from insertion of a point charge in the bulk of the support of eigenvalues. In particular, we characterize all rotationally symmetric scaling limits ('Mittag-Leffler fields') and obtain universality of them when the underlying potential is algebraic. Applications include a result on the asymptotic distribution of $\log|p_n(\zeta)|$ where $p_n$ is the characteristic polynomial of an $n$:th order random normal matrix.