The integrated density of states of the random graph Laplacian

TL;DR

This paper investigates the spectral properties of the random graph Laplacian in the percolating phase, deriving an integral equation for the density of states of the infinite cluster and exploring the existence of mobility edges and discrete eigenvalues.

Contribution

It introduces a method to isolate the density of states of the percolating cluster and derives a nonlinear integral equation solved via population dynamics.

Findings

Derived an integral equation for the density of states of the percolating cluster.

Provided evidence for the existence of a mobility edge.

Indicated the presence of discrete eigenvalues across the spectrum.

Abstract

We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum.