Edge scaling limits for a family of non-Hermitian random matrix ensembles

TL;DR

This paper investigates the edge behavior of eigenvalues in a family of non-Hermitian random matrices interpolating between GUE and Ginibre ensembles, revealing a transition in eigenvalue statistics from Poisson to Airy processes.

Contribution

It establishes scaling limits at the spectral edge for these interpolating ensembles and characterizes the transition between different eigenvalue statistics.

Findings

Eigenvalue distribution forms an ellipse that collapses to a line as non-Hermiticity decreases.

A transition from Poisson to Airy point process occurs at a critical scaling.

Maximum eigenvalue distribution interpolates between Gumbel and Tracy-Widom distributions.

Abstract

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.