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Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

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Abstract

We consider the integrated density of states (IDS) ρ∞(λ) of random Hamiltonian Hω=−Δ+Vω, Vω being a random field on ℝd which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|−1ρ(λ, HΛ(ω)), Λ ⊂ ℝd, around the thermodynamic limit ρ∞(λ) is bounded from above by exp {−k|Λ|},k>0. In this case ρ∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ∞(λ) as λ → 0+ in the case where Vω is non-negative. More precisely we prove the following bound: ρ∞(λ)≦exp(−kλ−d/2) as λ → 0+ k>0. This last result is then discussed in some examples.

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Authors and Affiliations

  1. Institut für Mathematik, Ruhr-Universität Bochum, Postfach 102148, D-4630, Bochum 1, Federal Republic of Germany

    Werner Kirsch & Fabio Martinelli

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  1. Werner Kirsch
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  2. Fabio Martinelli
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Communicated by B. Simon

On leave of absence from Istituto di Fisica, Università di Roma, ITALY, G.N.F.M. C.N.R

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Kirsch, W., Martinelli, F. Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians. Commun.Math. Phys. 89, 27–40 (1983). https://doi.org/10.1007/BF01219524

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  • DOI: https://doi.org/10.1007/BF01219524

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