Decorrelation in Local Statistics for random operators

TL;DR

This paper proves the complete independence of local spectral statistics at different energies in certain random operator models, extending previous partial results and confirming Poisson process behavior.

Contribution

It provides a full proof of independence of local spectral statistics at different energies for models where Minami's technique applies.

Findings

Local spectral statistics are Poisson processes with intensity proportional to the density of states.

Complete independence of local eigenvalue statistics at different energies is established.

Results apply to models where Minami's technique is valid.

Abstract

In this paper we study the local spectral statistics in the localised region of various random operator models, including the $d$-dimensional the Anderson model and random Schr\"odinger operators. It is already established, in the above models, that at an energy $E$, in the localised energy region of the spectrum, where the density of states $n(E) > 0$, the local eigenvalue statistics $X_E$ is a Poisson processes with intensity $n(E) \mathcal{L}$, $\mathcal{L}$ being the Lebesgue measure on $\mathbb{R}$. The question of independence of $X_E, X_{E^\prime}$ for distinct energies was partially solved in the literature. We solve it completely for all the models for which the Minami technique works.