Lifshitz tails on the Bethe lattice: a combinatorial approach

TL;DR

This paper investigates Lifshitz tails in the density of states for disordered models on the Bethe lattice, using a combinatorial approach to overcome challenges posed by exponential growth in these structures.

Contribution

It introduces a novel combinatorial method to analyze Lifshitz tails on the Bethe lattice, providing bounds on moments of the density of states.

Findings

Derived bounds on moments of the density of states.

Implications for the behavior of the integrated density of states.

Overcame difficulties due to exponential growth in lattice volume and surface.

Abstract

The density of states of disordered hopping models generically exhibits an essential singularity around the edges of its support, known as a Lifshitz tail. We study this phenomenon on the Bethe lattice, i.e. for the large-size limit of random regular graphs, converging locally to the infinite regular tree, for both diagonal and off-diagonal disorder. The exponential growth of the volume and surface of balls on these lattices is an obstacle for the techniques used to characterize the Lifshitz tails in the finite-dimensional case. We circumvent this difficulty by computing bounds on the moments of the density of states, and by deriving their implications on the behavior of the integrated density of states.