Poisson statistics of eigenvalues in the hierarchical Anderson model

TL;DR

This paper investigates the eigenvalue distribution in the hierarchical Anderson model, demonstrating Poisson fluctuations at any disorder level for models with spectral dimension less than one, using probabilistic methods.

Contribution

It provides a proof of Poisson eigenvalue statistics for the hierarchical Anderson model at arbitrary disorder when spectral dimension is below one, employing Minami's technique.

Findings

Poisson eigenvalue fluctuations for spectral dimension d<1

Applicable at arbitrary disorder levels

Elementary probabilistic proof provided

Abstract

We study the eigenvalue statistics for the hieracharchial Anderson model of Molchanov. We prove Poisson fluctuations at arbitrary disorder, when the the model has spectral dimension d<1. The proof is based on Minami's technique and we give an elementary exposition of the probabilistic arguments.