Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

TL;DR

This paper studies the spectral properties of a class of non-Hermitian matrices that are self-adjoint with respect to a certain inner product, revealing universal behaviors in their eigenvalue distributions that extend classical random matrix results.

Contribution

It introduces a new class of random matrices that are self-adjoint relative to a non-standard inner product and derives their universal spectral density and spacing distributions.

Findings

Level density is independent of the distribution of matrix entries.

Level-spacing distribution depends on the inner product and matrix symmetry, generalizing classical ensembles.

Results suggest new classes extending GOE and GUE based on the inner product.

Abstract

We investigate the level-density $\sigma(x)$ and level-spacing distribution $p(s)$ of random matrices $M=AF\neq M^{\dagger}$ where $F$ is a (diagonal) inner-product and $A$ is a random, real symmetric or complex Hermitian matrix with independent entries drawn from a probability distribution $q(x)$ with zero mean and finite higher moments. Although not Hermitian, the matrix $M$ is self-adjoint with respect to $F$ and thus has purely real eigenvalues. We find that the level density $\sigma_F(x)$ is independent of the underlying distribution $q(x)$, is solely characterized by $F$, and therefore generalizes Wigner's semicircle distribution $\sigma_W(x)$. We find that the level-spacing distributions $p(s)$ are independent of $q(x)$, are dependent upon the inner-product $F$ and whether $A$ is real or complex, and therefore generalize the Wigner's surmise for level spacing. Our results suggest $F$-dependent generalizations of the well-known Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) classes.