Spectral fluctuations for Schr\"odinger operators with a random decaying potential

TL;DR

This paper investigates the fluctuations of linear statistics for discrete Schrödinger operators with decaying randomness, identifying critical decay rates that influence the limit behavior of these fluctuations.

Contribution

It introduces a decomposition of polynomial spaces to determine how variance growth relates to decay rates, revealing phase transitions in fluctuation behavior.

Findings

Variance growth rates depend on polynomial subspaces

Critical decay exponents determine Gaussian or distribution-sensitive limits

Decomposition provides a framework for analyzing spectral fluctuation phases

Abstract

We study fluctuations of polynomial linear statistics for discrete Schr\"odinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.