Spectral fluctuations for the multi-dimensional Anderson model

TL;DR

This paper investigates the spectral fluctuations of the multi-dimensional Anderson model, demonstrating Gaussian convergence of polynomial linear statistics and classifying cases with zero variance.

Contribution

It provides a rigorous proof of Gaussian fluctuation convergence for the Anderson model and fully classifies rare potential cases with zero variance.

Findings

Fluctuations converge to Gaussian distribution.

Variance is positive for most potentials and polynomials.

Classifies rare zero-variance cases.

Abstract

In this paper, we examine fluctuations of polynomial linear statistics for the Anderson model on $\mathbb{Z}^d$ for any potential with finite moments. We prove that if normalized by the square root of the size of the truncated operator, these fluctuations converge to a Gaussian limit. For a vast majority of potentials and polynomials, we show that the variance of the limiting distribution is strictly positive, and we classify in full the rare cases in which this does not happen.