On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

TL;DR

This paper develops a supersymmetry-based method to analyze the joint probability density of eigenvalues and eigenvector overlaps in real and complex Ginibre ensembles, revealing heavy-tailed and finite-moment distributions in different regimes.

Contribution

It introduces a novel supersymmetry approach to derive explicit finite N distributions of eigenvalue and eigenvector overlaps in Ginibre ensembles, including edge and bulk scaling limits.

Findings

Real eigenvalues have heavy-tailed overlap distributions with divergent moments.

Complex eigenvalues have overlap distributions with finite first moments.

The method reproduces known results and extends analysis to new scaling regimes.

Abstract

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $\lambda$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue $z$ and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we provide the 'edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach \cite{BourgadeDubach}.