Analogies between random matrix ensembles and the one-component plasma in two-dimensions

TL;DR

This paper explores the analogy between eigenvalue distributions of various non-Hermitian random matrix ensembles and two-dimensional one-component plasma systems, revealing new insights into their density and correlation properties.

Contribution

It systematically identifies plasma systems corresponding to multiple random matrix ensembles across different element types, extending the plasma analogy to new classes and deriving related integral identities.

Findings

Eigenvalue densities follow from plasma analogy.

Eigenvalue correlations obey known plasma sum rules.

Derived integral identity for real quaternion matrices near the real axis.

Abstract

The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices $G$, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix $G_1^{-1} G_2$, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when each has either real, complex or real quaternion elements. One consequence of this analogy is that the leading form of the eigenvalue density follows as a corollary. Another is that the eigenvalue correlations must obey sum rules known to characterise the plasma system, and this leads us to a exhibit an integral identity satisfied by the two-particle correlation for real quaternion matrices in the neighbourhood of the real axis. Further random matrix ensembles investigated from this viewpoint are self dual non-Hermitian matrices, in which a previous study has related to the one-component plasma system in a disk at inverse temperature $\beta = 4$, and the ensemble formed by the single row and column of quaternion elements from a member of the circular symplectic ensemble.