Spectral properties of the Laplacian on bond-percolation graphs

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.Comment: 19 pages; presentation slightly improved, some comments and references added; to appear in Mathematische Zeitschrif

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

“…These results were complemented by Müller & Stollmann in a paper [83] where the percolating regime is studied. The present note continues the analysis [112] of site percolation Hamiltonians on general graphs and at the same time extends the results of [54] to bond percolation models on amenable Cayley graphs. In particular, part of our proofs relies on ideas introduced in [54] and extends them to a more general geometric setting.…”

“…Id, which can be show by direct calculation in the same manner as in [54]. Later we will see, that there is a more complete chain of inequalities between five operators.…”

“…The latter have a continuous IDS [28,24], while percolation Hamiltonians have a dense set of discontinuities, which survives even if one restricts the operator to infinite percolation clusters [50,18,112]. In [54] Kirsch & Müller analysed basic spectral features of bond percolation Laplacians on the lattice and moreover carried out a thorough study of the low-lying spectrum of these operators in the non-percolating regime. These results were complemented by Müller & Stollmann in a paper [83] where the percolating regime is studied.…”

An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters

“…subcritical percolation on the Cayley graphs of finitely generated groups are included as a special case. See [32,4] for estimates of the integrated density of states ('cumulative spectral measure') near the edges of the generator's spectrum in this case. However, also the critical [6], or supercritical percolation measure, conditioned on the finiteness of the cluster containing the root is a possible setting for the results proven, here.…”

“…The combinatorial Laplacian of subcritical Bernoulli-bond percolation graphs has been considered in [32] for Z d , and amenable graphs in [4]. For the adjacency matrix, as well as 'Laplacian spectra of graphs', interlacing is successfully applied (e.g.…”