Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs

TL;DR

This paper studies the spectral properties of Laplacians on supercritical bond-percolation graphs, revealing different asymptotic behaviors at spectral edges depending on boundary conditions, with implications for understanding complex network spectra.

Contribution

It provides a detailed analysis of the spectral asymptotics of Laplacians on percolation graphs, highlighting boundary condition effects and Lifshits and van Hove behaviors.

Findings

Lifshits tail at lower spectral edge for Dirichlet Laplacian

van Hove asymptotics for Neumann Laplacian at lower edge

Reversal of behaviors at the upper spectral edge

Abstract

We investigate Laplacians on supercritical bond-percolation graphs with different boundary conditions at cluster borders. The integrated density of states of the Dirichlet Laplacian is found to exhibit a Lifshits tail at the lower spectral edge, while that of the Neumann Laplacian shows a van Hove asymptotics, which results from the percolating cluster. At the upper spectral edge, the behaviour is reversed.