Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials

TL;DR

This paper studies the extreme eigenvalues of large discrete Anderson Hamiltonians with correlated Gaussian noise, revealing their convergence to a Poisson process and detailing the localization of eigenfunctions.

Contribution

It provides a detailed analysis of the spectral edge of correlated Gaussian Anderson Hamiltonians, including the convergence of eigenvalues and the localization properties of eigenfunctions.

Findings

Largest eigenvalues converge to a Poisson point process

Eigenfunctions localize near maxima of the noise

Relationship between eigenfunctions and noise maxima depends on covariance behavior

Abstract

We investigate the top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian noise in the large volume limit. The class of Gaussian noises under consideration allows for long-range correlations. We show that the largest eigenvalues converge to a Poisson point process and we obtain a very precise description of the associated eigenfunctions near their localisation centres. We also relate these localisation centres with the locations of the maxima of the noise. Actually, our analysis reveals that this relationship depends in a subtle way on the behaviour near $0$ of the covariance function of the noise: in some situations, the largest eigenfunctions are not associated with the largest values of the noise.