Spectral Analysis of Percolation Hamiltonians

TL;DR

This paper investigates the spectral properties of percolation Hamiltonians, characterizing eigenvalues with finitely supported eigenfunctions, and analyzing the behavior of the integrated density of states, including at points of discontinuity, with applications to mixed models.

Contribution

It provides a detailed spectral analysis of percolation Hamiltonians, characterizes the eigenvalues with finitely supported eigenfunctions, and extends results to mixed Anderson-Quantum percolation models.

Findings

Eigenvalues with finitely supported eigenfunctions form a dense subset of algebraic integers.

The integrated density of states has discontinuities exactly at these eigenvalues.

Convergence of the integrated densities of states holds even at points of discontinuity.

Abstract

We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard.