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Lifshitz Tails on the Bethe Lattice: A Combinatorial Approach

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Abstract

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.

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Authors and Affiliations

  1. LPTENS, Unité Mixte de Recherche (UMR 8549) du CNRS et de l’ENS, associée à l’UPMC, Univ. Paris 06, 24 Rue Lhomond, 75231, Paris Cedex 05, France

    Victor Bapst & Guilhem Semerjian

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  1. Victor Bapst
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  2. Guilhem Semerjian
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Correspondence to Victor Bapst.

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Bapst, V., Semerjian, G. Lifshitz Tails on the Bethe Lattice: A Combinatorial Approach. J Stat Phys 145, 51 (2011). https://doi.org/10.1007/s10955-011-0319-3

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  • DOI: https://doi.org/10.1007/s10955-011-0319-3

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