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The Thouless formula for random non-Hermitian Jacobi matrices

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Abstract

Random non-Hermitian Jacobi matricesJ n of increasing dimensionn are considered. We prove that the normalized eigenvalue counting measure ofJ n converges weakly to a limiting measure μ asn→∞. We also extend to the non-Hermitian case the Thouless formula relating μ and the Lyapunov exponent of the second-order difference equation associated with the sequenceJ n . The measure μ is shown to be log-Hölder continuous. Our proofs make use of (i) the theory of products of random matrices in the form first offered by H. Furstenberg and H. Kesten in 1960 [8], and (ii) some potential theory arguments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, U.K.

    Ilya Ya. Goldsheid & Boris A. Khoruzhenko

  2. Department of Mathematical Sciences, Brunel University, UB8 3PH, London, U.K.

    Boris A. Khoruzhenko

Authors
  1. Ilya Ya. Goldsheid
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  2. Boris A. Khoruzhenko
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Correspondence to Ilya Ya. Goldsheid.

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Goldsheid, I.Y., Khoruzhenko, B.A. The Thouless formula for random non-Hermitian Jacobi matrices. Isr. J. Math. 148, 331–346 (2005). https://doi.org/10.1007/BF02775442

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  • DOI: https://doi.org/10.1007/BF02775442

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