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.

19 pages; presentation slightly improved, some comments and references added; to appear in Mathematische Zeitschrift

References in corpus (4)

• Spectral Analysis of Percolation Hamiltonians
• Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs
• Quantum site percolation on amenable graphs
• Critical behaviour of the Rouse model for gelling polymers

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