Lifshitz tails on the Bethe lattice: a combinatorial approach

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.

32 pages, 10 figures; improved results on the asymptotic behavior of the integrated density of states

References in corpus (5)

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• On the localization transition in symmetric random matrices
• An Invitation to Random Schroedinger operators
• Extended States in a Lifshitz Tail Regime for Random Schrödinger Operators on Trees
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Cited by in corpus (12)

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• The Large Connectivity Limit of the Anderson Model on Tree Graphs
• Spectral properties of the trap model on sparse networks
• Index statistical properties of sparse random graphs
• Surprises in the phase diagram of the Anderson model on the Bethe lattice
• Calculation of mean spectral density for statistically uniform tree-like random models
• Eigenvalue spectral tails and localization properties of asymmetric networks
• Statistical mechanics of vector Hopfield network near and above saturation
• The umpteen operator and its Lifshitz tails
• Disordered harmonic chain with random masses and springs: a combinatorial approach