Israel Journal of Mathematics

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

The Thouless formula for random non-Hermitian Jacobi matrices



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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Avron and B. Simon,Almost periodic Schrödinger operators II. The integrated density of states, Duke Mathematical Journal50 (1983), 369–391.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. Bougerol and J. Lacroix,Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics., Vol. 8, Birkhäuser, Boston-Basel-Stuttgart, 1985.MATHGoogle Scholar
  3. [3]
    R. Carmona and J. Lacroix,Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.MATHGoogle Scholar
  4. [4]
    W. Craig and B. Simon,Subharmonicity of the Lyapunov Index, Duke Mathematical Journal50 (1983), 551–560.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    B. Derrida, J. L. Jacobsen and R. Zeitak,Lyapunov exponent and density of states of a one-dimensional non-Hermitian Schrodinger equation, Journal of Statistical Physics98 (2000), 31–55.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    A. L. Figotin and L. A. Pastur,The positivity of the Lyapunov exponent and the absence of the absolutely continuous spectrum for almost-Mathieu equation, Journal of Mathematical Physics25 (1984), 774–777.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Furstenberg,Noncommuting random products, Transactions of the American Mathematical Society108 (1963), 377–428.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    H. Furstenberg and H. Kesten,Products of random matrices, Annals of Mathematical Statistics,31 (1960), 457–469.CrossRefMathSciNetMATHGoogle Scholar
  9. [9]
    I. C. Gohberg and M. G. Krein,Introduction to the Theory of Linear Nonself-adjoint Operators, American Mathematical Society, Providence, RI, 1969.Google Scholar
  10. [10]
    I. Ya. Goldsheid and B. A. Khoruzhenko,Eigenvalue curves of asymmetric tridiagonal random matrices, Electronic Journal of Probability5 (2000), Paper 16, 26 pp.Google Scholar
  11. [11]
    I. Ya. Goldsheid and B. A. Khoruzhenko,Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model, Communications in Mathematical Physics,238 (2003), 505–524.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Y. Guivarc'h and, A. Raugi,Frontière de Furstenberg properiétés de contraction et théorèmes de convergence, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete69 (1985), 187–242.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    N. Hatano and D. R. Nelson,Localization transitions in non-Hermitian quantum mechanics, Physical Review Letters77 (1996), 570–573.CrossRefGoogle Scholar
  14. [14]
    N. Hatano and D. R. Nelson,Vortex pinning and non-Hermitian quantum mechanics, Physical ReviewB56 (1997), 8651–8673.CrossRefGoogle Scholar
  15. [15]
    D. E. Holz, H. Orland and A. Zee,On the remarkable spectrum of a non-Hermitian random matrix model, Journal of Physics. A. Mathematical and General36 (2003), 3385–3400.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    L. Hörmander,The Analysis of Linear Partial Differential Equations, Vol. I, Springer, Berlin, 1983.Google Scholar
  17. [17]
    L. Hörmander,Notions of Convexity, Birkhäuser, Boston, 1994.MATHGoogle Scholar
  18. [18]
    V. I. Oseledec,A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Transactions of the Moscow Mathematical Society19 (1968), 197–221.MathSciNetGoogle Scholar
  19. [19]
    L. A. Pastur and A. L. Figotin,Spectra of Random and Almost-periodic Operators, Springer, Berlin-Heidelberg-New York, 1992.MATHGoogle Scholar
  20. [20]
    A. D. Virtser,On products of random matrices and operators (English), Theory of Probability and its Applications24 (1980), 367–377.MATHCrossRefGoogle Scholar
  21. [21]
    H. Widom,Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of the Toeplitz determinants in the case of nonvanishing index, Operator Theory: Advances and Applications48 (1990), 387–421.MathSciNetGoogle Scholar
  22. [22]
    H. Widom,Eigenvalue distribution for nonselfadjoint Toeplitz matrices, Operator Theory: Advances and Applications71 (1994), 1–8.MathSciNetGoogle Scholar

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.

Personalised recommendations