Uniform approximation of the integrated density of states for long-range percolation Hamiltonians

TL;DR

This paper demonstrates that the integrated density of states for long-range percolation Hamiltonians on amenable groups can be uniformly approximated, enabling the characterization of its discontinuities.

Contribution

It introduces a method for uniform approximation of the IDS for long-range percolation graphs on amenable groups, advancing spectral analysis techniques.

Findings

Uniform approximation of the IDS established

Characterization of the discontinuities of the IDS

Application to long-range percolation Hamiltonians

Abstract

In this paper we study the spectrum of long-range percolation graphs. The underlying geometry is given in terms of a finitely generated amenable group. We prove that the integrated density of states (IDS) or spectral distribution function can be approximated uniformly in the energy variable. Using this, we are able to characterise the set of discontinuities of the IDS.