Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

TL;DR

This paper investigates how a hard wall boundary influences the large N asymptotic expansion of the partition function in 2D Coulomb gases with radially symmetric potentials, revealing new universal constants and modified logarithmic terms.

Contribution

It provides the first detailed analysis of the impact of an internal hard wall on the asymptotics of Coulomb gas partition functions in the radially symmetric case.

Findings

The order log N term's coefficient changes with the hard wall position.

Universal constants are identified at order √N and constant terms.

Different rational prefactors appear for the log N term depending on the wall location.

Abstract

The large $N$ asymptotic expansion of the partition function for the normal matrix model is predicted to have special features inherited from its interpretation as a two-dimensional Coulomb gas. However for the latter, it is most natural to include a hard wall at the boundary of the droplet. We probe how this affects the asymptotic expansion in the solvable case that the potential is radially symmetric and the droplet is a disk or an annulus. We allow too for the hard wall to be strictly inside the boundary of the droplet. It is observed the term of order $\log N$, has then a different rational number prefactor to that when the hard wall is at the droplet boundary. Also found are certain universal (potential independent) numerical constants given by definite integrals, both at order $\sqrt{N}$, and in the constant term.