Lifshits tails in the hierarchical Anderson model

TL;DR

This paper proves that the hierarchical Anderson model shows Lifshits tails at the spectrum's edge, with the tail behavior characterized by the spectral dimension, using Dirichlet-Neumann bracketing and large deviations.

Contribution

It establishes the presence of Lifshits tails in the hierarchical Anderson model and relates the Lifshits exponent to the spectral dimension of the structure.

Findings

Lifshits tail at the spectrum's upper edge

Lifshits exponent linked to spectral dimension

Method based on Dirichlet-Neumann bracketing and large deviations

Abstract

We prove that the homogeneous hierarchical Anderson model exhibits a Lifshits tail at the upper edge of its spectrum. The Lifshits exponent is given in terms of the spectral dimension of the homogeneous hierarchical structure. Our approach is based on Dirichlet-Neumann bracketing for the hierarchical Laplacian and a large-deviation argument.