Edge behavior of two-dimensional Coulomb gases near a hard wall

TL;DR

This paper studies the behavior of two-dimensional Coulomb gases near a hard wall, revealing singular density components and analyzing local particle correlations in the determinantal case, with connections to truncated unitary matrices.

Contribution

It introduces a detailed analysis of Coulomb gas density singularities at a hard wall and characterizes local correlations using Laplace-type integrals in the determinantal setting.

Findings

Density acquires a singular component at the hard wall.

Local correlations are expressed via Laplace-type integrals.

Connections to truncated unitary matrices in weak non-unitarity regime.

Abstract

We consider a two-dimensional Coulomb gas confined to a disk when the external potential is radially symmetric. In the presence of a hard-wall constraint effective to change the equilibrium, the density of the equilibrium measure acquires a singular component at the hard wall. In the determinantal case, we study the local statistics of Coulomb particles at the hard wall and prove that their local correlations are expressed in terms of "Laplace-type" integrals, which appear in the context of truncated unitary matrices in the regime of weak non-unitarity.