Minami's estimate: beyond rank one perturbation and monotonicity

TL;DR

This paper extends Minami's estimate to certain alloy-type models with sign-changing potentials and uses it to establish Poisson statistics for energy level spacings, broadening understanding of spectral properties in disordered systems.

Contribution

The paper generalizes Minami's estimate to models with sign-changing potentials, enabling analysis of spectral statistics in more complex disordered systems.

Findings

Minami's estimate holds for a class of alloy-type models with sign-changing potentials.

Poisson statistics are proven for energy level spacings in these models.

Results apply to potentials close to the standard Anderson model.

Abstract

In this note we prove Minami's estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami's estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).