Poisson statistics for random deformed band matrices with power law band width

TL;DR

This paper demonstrates Poisson statistics for certain random band matrices with Gaussian entries, including cases with very small components, and provides bounds on the density of states for complex deformed matrices.

Contribution

It establishes Poisson statistics for Gaussian band matrices with minimal entry sizes and derives uniform upper and lower bounds on the density of states.

Findings

Poisson statistics hold for band matrices with Gaussian entries as small as n^{- ext{epsilon}}.

Applicable to band matrices cut from GUE with width satisfying w^{3.5} << n.

Provides bounds on the density of states for complex deformed Gaussian band matrices.

Abstract

We show Poisson statistics for random band matrices which diagonal entries have Gaussian components. These components are possibly as small as $n^{-\varepsilon}$. Particularly, our result is applicable for a band matrix cut from the GUE with the band width satisfying $w^{3.5}<<n$. A uniform upper bound of the averaged density of states (DOS) is obtained for complex deformed Gaussian band matrices with arbitrary $w$. A lower estimate of the DOS is also proven for arbitrary $w$ in a certain class of band matrices.