Spectral properties of the Laplacian on bond-percolation graphs

TL;DR

This paper investigates the spectral characteristics of Laplacians on bond-percolation graphs, revealing how their spectra and density of states behave, especially near spectral edges, with implications for understanding disorder in random graphs.

Contribution

It provides a detailed analysis of the spectral properties and Lifshits tails of Laplacians on bond-percolation graphs, highlighting dimension-dependent and independent behaviors.

Findings

Spectrum is almost surely [0,4d]

Lifshits tails occur at spectral edges in the non-percolating phase

Characteristic exponents depend on spectral edge and dimension

Abstract

Bond-percolation graphs are random subgraphs of the d-dimensional integer lattice generated by a standard bond-percolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, self-adjoint, ergodic random operators with off-diagonal disorder. They possess almost surely the non-random spectrum [0,4d] and a self-averaging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the non-percolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge, where the exponent equals 1/2.