Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $\beta$

TL;DR

This paper introduces a random matrix model for the 2D Coulomb gas at various temperatures, deriving an analytic spacing distribution that fits numerical data and applies to different physical and mathematical systems.

Contribution

It proposes a novel 2D Coulomb gas representation for non-Hermitian matrices and introduces an effective 2_{ m eff}(eta) to accurately model eigenvalue spacings.

Findings

Analytic spacing distribution matches numerical data for small 2.

The model reproduces the Poisson distribution at 2=0.

Fits eigenvalue spacing data from quantum and ecological systems.

Abstract

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature $\beta$. For $2\times 2$ matrices with Gaussian distribution we analytically compute the nearest neighbour spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective $\beta_{\rm eff}(\beta)$ in our analytic formula, that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of $\beta$. It reproduces the 2D Poisson distribution at $\beta=0$ exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by non-integer values $\beta=1.4$ and $\beta=2.6$ from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.