Finite-rank complex deformations of random band matrices: sigma-model approximation

TL;DR

This paper investigates the eigenvalue distribution of large complex deformed random band matrices, showing that under certain conditions, their spectral density aligns with that of the Gaussian Unitary Ensemble using sigma-model approximation techniques.

Contribution

It introduces a sigma-model approximation approach to analyze finite-rank complex deformations of random band matrices, extending understanding of their spectral properties.

Findings

Eigenvalue density matches GUE in the limit

Band width W must grow faster than sqrt(N)

Method extends previous sigma-model techniques

Abstract

We study the distribution of complex eigenvalues $z_1,\ldots, z_N$ of random Hermitian $N\times N$ block band matrices with a complex deformation of a finite rank. Assuming that the width of the band $W$ grows faster than $\sqrt{N}$, we proved that the limiting density of $\Im z_1,\ldots, \Im z_N$ in a sigma-model approximation coincides with that for the Gaussian Unitary Ensemble. The method follows the techniques of arXiv:1802.03813