The real Ginibre ensemble with $k = O(n)$ real eigenvalues

TL;DR

This paper analyzes the spectral properties of the real Ginibre ensemble with a linear number of real eigenvalues, establishing large deviations principles and asymptotic probabilities for the eigenvalue distribution.

Contribution

It introduces a two-phase log-gas model and derives asymptotic expansions for the probability of having a specified fraction of real eigenvalues.

Findings

Large deviations principle for eigenvalue density

Asymptotic expansion for probability of real eigenvalues

Characterization of spectral measures

Abstract

We consider the ensemble of Real Ginibre matrices with a positive fraction $\alpha>0$ of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and we introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability $p^n_{\alpha n}$ that an $n\times n$ Ginibre matrix has $k=\alpha n$ real eigenvalues and we characterize the spectral measures of these matrices.