The complex elliptic Ginibre ensemble at weak non-Hermiticity: bulk spacing distributions

TL;DR

This paper investigates the eigenvalue spacing distribution in the complex elliptic Ginibre ensemble at weak non-Hermiticity, revealing a crossover from GUE to Poisson statistics described by a generalized Gaudin-Mehta distribution involving Painlevé functions.

Contribution

It introduces a generalized Gaudin-Mehta distribution for bulk spacings in the elliptic Ginibre ensemble at weak non-Hermiticity, connecting known statistics across different regimes.

Findings

Distribution converges to a generalized Gaudin-Mehta form

Describes crossover from GUE to Poisson statistics

Involves an integro-differential Painlevé function

Abstract

We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlev\'e function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap statistics for eigenvalue real parts in the bulk of the Complex Ginibre Ensemble.