Mean eigenvector self-overlap in deformed complex Ginibre ensemble

TL;DR

This paper studies the microscopic behavior of mean eigenvector overlaps in a deformed complex Ginibre ensemble, revealing phase transitions and dependence on the deformation's eigenvalues.

Contribution

It characterizes the microscopic statistics of eigenvector overlaps near the edge and for outliers, introducing iterative erfc integrals depending on the deformation's eigenvalues.

Findings

Microscopic statistics near the edge are described by iterative erfc integrals.

Phase transition phenomenon for mean diagonal overlap depending on eigenvalue modulus.

Dependence of overlap statistics on the geometric multiplicity of eigenvalues of the deformation.

Abstract

Consider a random matrix of size $N$ as an additive deformation of the complex Ginibre ensemble under a deterministic matrix $X_0$ with a finite rank, independent of $N$. We prove that microscopic statistics for the mean diagonal overlap, near the edge point, are characterized by the iterative erfc integrals, which only depend on the geometric multiplicity of certain eigenvalue of $X_0$. We also investigate the microscopic statistics for the mean diagonal overlap of the outlier eigenvalues. Further we get a phenomenon of the phase transition for the mean diagonal overlap, with respect to the modulus of the eigenvalues of $X_0$.