Spectral density of complex eigenvalues and associated mean eigenvector self-overlaps at the edge of elliptic Ginibre ensembles

TL;DR

This paper investigates the spectral density and eigenvector overlaps at the edge of elliptic Ginibre ensembles, revealing universal behavior in strong non-Hermiticity and ensemble-dependent results in weak non-Hermiticity.

Contribution

It provides a detailed analysis of eigenvalue density and eigenvector overlaps at the spectral edge, highlighting universality in strong non-Hermiticity and differences in weak non-Hermiticity regimes.

Findings

Universal behavior of eigenvalue density and overlaps at strong non-Hermiticity

Different results for density and overlaps across ensembles in weak non-Hermiticity

Completes a series of studies on eigenvector self-overlaps in elliptic Ginibre ensembles

Abstract

We consider the density of complex eigenvalues, $\rho(z)$, and the associated mean eigenvector self-overlaps, $\mathcal{O}(z)$, at the spectral edge of $N \times N$ real and complex elliptic Ginibre matrices, as $N \to \infty$. Two different regimes of ellipticity are studied: strong non-Hermiticity, keeping the ellipticity parameter $\tau$ fixed and weak non-Hermiticity with $\tau \rightarrow 1 $ as $N \rightarrow \infty$. At strong non-Hermiticity, we find that both $\rho(z)$ and $\mathcal{O}(z)$ have the same leading order behaviour across the elliptic Ginibre ensembles, establishing the expected universality. In the limit of weak non-Hermiticity, we find different results for $\rho(z)$ and $\mathcal{O}(z)$ across the two ensembles. This paper is the final of three papers that we have presented addressing the mean self-overlap of eigenvectors in these ensembles.