Limit Theorems on the Mesoscopic Scale for the Anderson Model

TL;DR

This paper establishes a central limit theorem for eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale within a regime of exponential localization.

Contribution

It provides the first proof of a central limit theorem for eigenvalue counting functions in the mesoscopic scale for the Anderson model.

Findings

Eigenvalue fluctuations follow a normal distribution in the mesoscopic scale.

The result holds in the regime of exponential localization.

The study advances understanding of spectral properties in disordered quantum systems.

Abstract

In this paper, we study eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale. We carry out this study in a regime of exponential localization and prove a central limit theorem for the eigenvalue counting function in a shrinking interval.