Mean left-right eigenvector self-overlap in the real Ginibre ensemble

TL;DR

This paper analytically investigates the mean eigenvector overlap in the real Ginibre ensemble, revealing similarities with the complex Ginibre ensemble in bulk and edge regions, but notable differences near the real axis.

Contribution

It derives a finite N expression for the mean eigenvector overlap in the real Ginibre ensemble and explores its asymptotic behavior across different spectral regions.

Findings

Limiting overlaps match complex Ginibre results in bulk and edge

Distinct asymptotics near the real axis depletion region

Numerical evidence suggests similar overlap distributions in bulk and edge for both ensembles

Abstract

We study analytically the Chalker-Mehlig mean diagonal overlap $\mathcal{O}(z)$ between left and right eigenvectors associated with a complex eigenvalue $z$ of $N\times N$ matrices in the real Ginibre ensemble (GinOE). We first derive a general finite $N$ expression for the mean overlap and then investigate several scaling regimes in the limit $N\rightarrow \infty$. While in the generic spectral bulk and edge of the GinOE the limiting expressions for $\mathcal{O}(z)$ are found to coincide with the known results for the complex Ginibre ensemble (GinUE), in the region of eigenvalue depletion close to the real axis the asymptotic for the GinOE is considerably different. We also study numerically the distribution of diagonal overlaps and conjecture that it is the same in the bulk and at the edge of both the GinOE and GinUE, but essentially different in the depletion region of the GinOE.