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Mathematics  2013 

Lifshitz asymptotics for percolation Hamiltonians

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Abstract:

We study a discrete Laplace operator $\Delta$ on percolation subgraphs of an infinite graph. The ball volume is assumed to grow at most polynomially. We are interested in the behavior of the integrated density of states near the lower spectral edge. If the graph is a Cayley graph we prove that it exhibits Lifshitz tails. If we merely assume that the graph has an exhausting sequence with positive $\delta$-dimensional density, we obtain an upper bound on the integrated density of states of Lifshitz type.

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