Real eigenvalues of elliptic random matrices

TL;DR

This paper studies the distribution and statistics of real eigenvalues in large elliptic Ginibre matrices with correlated entries, revealing their limiting behavior and interpolating distributions between classical laws.

Contribution

It provides the first large-$N$ expansion for the mean and variance of real eigenvalues in elliptic matrices with non-Hermiticity parameter scaling as $N^{-1}$.

Findings

Derived the large-$N$ expansion of mean and variance of real eigenvalues.

Established the limiting empirical distribution interpolating Wigner semicircle and uniform law.

Used skew-orthogonal polynomial techniques for correlation kernel analysis.

Abstract

We consider the real eigenvalues of an $(N \times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau_N\in [0,1]$. In the almost-Hermitian regime where $1-\tau_N=\Theta(N^{-1})$, we obtain the large-$N$ expansion of the mean and the variance of the number of the real eigenvalues. Furthermore, we derive the limiting empirical distributions of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution, the restriction of the elliptic law on the real axis. Our proofs are based on the skew-orthogonal polynomial representation of the correlation kernel due to Forrester and Nagao.