Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics

TL;DR

This paper investigates the transition in eigenvalue statistics of random matrices from Hermitian to non-Hermitian regimes, revealing unique behaviors such as a non-standard nearest neighbor distribution.

Contribution

It introduces a method to analyze the crossover of eigenvalue statistics using orthogonal polynomials, highlighting novel spectral behaviors in the transition.

Findings

Eigenvalue statistics transition from Wigner-Dyson to Ginibre.

Identification of unusual $p(s) \\propto s^{5/2}$ behavior.

Detailed analysis of spectral form factor and number variance.

Abstract

By using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner- Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures (as e.g. spectral form factor, number variance and small distance behavior of the nearest neighbor distance distribution $p(s)$) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter values.