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.
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Peter Müller: - On leave from: Institut für Theoretische Physik, Georg-August-Universität, Friedrich-Hund-Platz 1, D–37077 Göttingen, Germany
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Kirsch, W., Müller, P. Spectral properties of the Laplacian on bond-percolation graphs. Math. Z. 252, 899–916 (2006). https://doi.org/10.1007/s00209-005-0895-5
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DOI: https://doi.org/10.1007/s00209-005-0895-5