Poisson Statistics in the Non-Homogeneous Hierarchical Anderson Model

TL;DR

This paper investigates eigenvalue localization in a non-homogeneous hierarchical Anderson model, establishing conditions for eigenvalue distribution convergence to a Poisson process, and explores how spectral dimension influences disorder effects.

Contribution

It extends existing methods to non-i.i.d. potentials and analyzes the impact of spectral dimension on eigenvalue statistics in hierarchical models.

Findings

Eigenvalues converge to a Poisson point process under certain conditions.

Disorder magnitude affects eigenvalue distribution and spectral properties.

Spectral dimension influences the relationship between disorder and localization.

Abstract

In this article we study the problem of localization of eigenvalues for the non-homogeneous hierarchical Anderson model. More specifically, given the hierarchical Anderson model with spectral dimension $0<d<1$ with a random potential acting on the diagonal of non i.i.d. random variables, sufficient conditions on the disorder are provided in order to obtain the two main results: the weak convergence of the counting measure for almost all realization of the random potential and the weak convergence of the re-scaled eigenvalue counting measure to a Poisson point process. The technical part improves the already existing arguments of Kritchevski , who studied the hierarchical model with a disorder acting on the diagonal, with independent and identically distributed random variables, by using the argument of Minami . At the end of this article, we study an application example that allows us to understand some relations between the spectral dimension of the hierarchical Laplacian and the magnitude of the disorder.