Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices

TL;DR

This paper derives explicit formulas for eigenvalue and eigenvector statistics in complex symmetric random matrices, revealing unique edge behaviors that differ from classical Ginibre ensembles and suggesting universality across different distributions.

Contribution

It provides the first explicit joint distribution of eigenvalues and eigenvectors for the class AI$^ ext{d}$ of complex symmetric matrices, extending understanding beyond Ginibre ensembles.

Findings

Eigenvalue density and eigenvector overlaps are derived for finite and large N.

Edge behavior of spectral statistics differs from Ginibre universality.

Numerical evidence supports universality across different matrix distributions.

Abstract

Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^\dag$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices.