Aron Wennman: A direct approach to local scaling limits for determinantal 2D Coulomb gases

Abstract

The 2D Coulomb gas is a model of charged particles in the plane, and at inverse temperature β=2 it describes the eigenvalue distributions of natural non-Hermitian random matrices. These point processes are determinantal: their statistics are encoded by certain weighted Bergman kernels for polynomial subspaces.

Traditionally, one studies these kernels through the associated orthogonal polynomials. But in many interesting situations, such as when the gas is confined by hard walls or when the equilibrium support has several connected components, the orthogonal polynomial method becomes challenging.

I will describe a new approach that avoids orthogonal polynomials entirely. Instead, it works directly with a Hilbert space of entire functions, obtained by rescaling near a given point. We can then study these Hilbert spaces using classical ideas from Paley–Wiener theory and potential theory. This direct approach gives several new universality results showing that, under some regularity assumptions, the microscopic behaviour of the gas only depends on the local properties of its equilibrium measure.

Based on ongoing joint work with Joakim Cronvall.