Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs

TL;DR

This paper proves the uniform existence of the integrated density of states for random Hamiltonians on long-range percolation graphs, using a trace formula, advancing understanding of spectral properties in disordered systems.

Contribution

It establishes the uniform existence of the IDS for a class of random Hamiltonians on long-range percolation graphs and provides a trace formula representation.

Findings

Proves uniform existence of the IDS for these Hamiltonians

Expresses the IDS via a Pastur-Shubin trace formula

Extends spectral analysis to long-range percolation graph models

Abstract

In this paper we consider random Hamiltonians defined on long-range percolation graphs over $\ZZ^d$. The Hamiltonian consists of a randomly weighted Laplacian plus a random potential. We prove uniform existence of the integrated density of states and express the IDS using a Pastur-Shubin trace formula.