The integrated density of states of the random graph Laplacian

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.