Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations

TL;DR

This paper analyzes the eigenvalue distribution of large bi-diagonal Toeplitz matrices under small Gaussian perturbations, providing precise asymptotic descriptions of the average eigenvalue density within the convex hull of the symbol's range.

Contribution

It offers a detailed asymptotic characterization of eigenvalue density for perturbed bi-diagonal matrices, advancing understanding of spectral behavior under randomness.

Findings

Derived asymptotic eigenvalue density formulas

Identified eigenvalue distribution within the convex hull

Enhanced spectral analysis of perturbed Toeplitz matrices

Abstract

We study the spectrum of large a bi-diagonal Toeplitz matrix subject to a Gaussian random perturbation with a small coupling constant. We obtain a precise asymptotic description of the average density of eigenvalues in the interior of the convex hull of the range symbol.