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Lifshitz tails on the Bethe lattice: a combinatorial approach

By Victor Bapst and Guilhem Semerjian


The density of states of disordered hopping models generically exhibits an essential singularity around the edges of its support, known as a Lifshitz tail. We study this phenomenon on the Bethe lattice, i.e. for the large-size limit of random regular graphs, converging locally to the infinite regular tree, for both diagonal and off-diagonal disorder. The exponential growth of the volume and surface of balls on these lattices is an obstacle for the techniques used to characterize the Lifshitz tails in the finite-dimensional case. We circumvent this difficulty by computing bounds on the moments of the density of states, and by deriving their implications on the behavior of the integrated density of states.Comment: 32 pages, 10 figures; improved results on the asymptotic behavior of the integrated density of state

Topics: Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Statistical Mechanics, Mathematical Physics, Mathematics - Probability
Year: 2011
DOI identifier: 10.1007/s10955-011-0319-3
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