Microscopic densities and Fock-Sobolev spaces

TL;DR

This paper investigates microscopic densities near singular points in eigenvalue ensembles, establishing their convergence to equilibrium densities and deriving asymptotics for Bergman functions in Fock-Sobolev spaces.

Contribution

It provides new results on microscopic density behavior near singularities and connects these to asymptotics of Bergman functions in Fock-Sobolev spaces.

Findings

Microscopic densities rapidly approach classical equilibrium densities away from singularities.

Asymptotic formulas for Bergman functions in specific Fock-Sobolev spaces.

Analysis of eigenvalue ensembles near bulk singular points.

Abstract

We study two-dimensional eigenvalue ensembles close to certain types of singular points in the bulk of the droplet. We prove existence of a microscopic density which quickly approaches the classical equilibrium density, as the distance from the singularity increases beyond the microscopic scale. As a consequence we obtain asymptotics for the Bergman function of certain Fock-Sobolev spaces of entire functions.