Lifshitz asymptotics for percolation Hamiltonians

TL;DR

This paper investigates the spectral properties of percolation Hamiltonians on infinite graphs, establishing Lifshitz tail behavior of the integrated density of states under certain geometric conditions.

Contribution

It proves Lifshitz asymptotics for the integrated density of states on Cayley graphs and provides upper bounds for more general graphs with positive density.

Findings

Lifshitz tails are proven for Cayley graphs.

Upper bounds on the density of states are established for graphs with positive density.

The results connect geometric properties of graphs to spectral asymptotics.

Abstract

We study a discrete Laplace operator $\Delta$ on percolation subgraphs of an infinite graph. The ball volume is assumed to grow at most polynomially. We are interested in the behavior of the integrated density of states near the lower spectral edge. If the graph is a Cayley graph we prove that it exhibits Lifshitz tails. If we merely assume that the graph has an exhausting sequence with positive $\delta$-dimensional density, we obtain an upper bound on the integrated density of states of Lifshitz type.