Spectral properties of the Laplacian on bond-percolation graphs |
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Authors: | Werner Kirsch Peter Müller |
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Affiliation: | 1. Fakult?t und Institut für Mathematik, Ruhr-Universit?t Bochum, Universit?tsstra?e 150, D–44780, Bochum, Germany
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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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Keywords: | 47B80 34B45 05C80 |
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