Israel Journal of Mathematics

, Volume 148, Issue 1, pp 331–346 | Cite as

The Thouless formula for random non-Hermitian Jacobi matrices

  • Ilya Ya. GoldsheidEmail author
  • Boris A. Khoruzhenko


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.


Lyapunov Exponent Random Matrice Eigenvalue Distribution Schr6dinger Operator Hermitian Case 
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Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Queen MaryUniversity of LondonLondonU.K.
  2. 2.Department of Mathematical SciencesBrunel UniversityLondonU.K.

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