# Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

## TL;DR

This paper analyzes a family of two-dimensional Coulomb gases confined to an ellipse, deriving explicit correlation functions at inverse temperature 2 using orthogonal polynomials, and explores universality classes in large particle limits.

## Contribution

It provides explicit formulas for correlation functions of Coulomb gases on an ellipse using orthogonal polynomials and investigates their universality in large-$N$ limits.

## Key findings

- Explicit correlation functions via Gegenbauer and Jacobi polynomials.
- Identification of new universality classes in large-$N$ limits.
- Recovery of sine- and Bessel-kernels in Hermitian limit.

## Abstract

We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse, due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature $\beta=2$ we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all $k$-point correlation functions for a fixed number of particles $N$. These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-$N$ limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine- and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.