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On large deviations for the spectral measure of discrete Coulomb gas

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1934)

Abstract

We establish a large deviation principle for the spectral measure of a large class of discrete Coulomb gas. The setting includes invariant ensembles from the classical orthogonal polynomials which are the discrete analogues of the continuous random matrix models. The proof requires a refinement of the arguments used in the continuous framework due to the constraint that may appear in the description of the rate functional. Our analysis closely follows the investigations of K. Johansson at the level of the largest eigenvalue, that is recovered here by a change of variables.

Key words

  • large deviations
  • discrete Coulomb gas
  • spectral measure
  • largest eigenvalue
  • random matrix models
  • continuous Coulomb gas

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References

  • [Bai] Z. Bai, Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist. Sinica 9, 611–677 (1999).

    MATH  MathSciNet  Google Scholar 

  • [BKMM] J. Baik, T. Kriecherbauer, K.T.-R. McLaughlin and P.D. Miller, Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles, Arxiv math. CA/0310278 (2003).

    Google Scholar 

  • [BDG] G. Ben Arous, A. Dembo and A. Guionnet, Aging of Spherical Spin Glasses, Probab. Theory Relat. Fields 120, 1–67 (2001).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [B-G] G. Ben Arous, and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s Non-Commutative Entropy, Probab. Theory Relat. Fields 108, 517–542 (1997).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [De-Ze] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Springer-Verlag, (1998).

    Google Scholar 

  • [De-St] J.D. Deuschel and D.W. Stroock, Large deviations, Academic Press-Boston, (1989).

    MATH  Google Scholar 

  • [Dr-Sa1] P.D. Dragnev and E.B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72, 223–259 (1997).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [Dr-Sa2] P.D. Dragnev and E.B. Saff, A problem in potential theory and zero asymptotics of Krawtchouk polynomials, Journal of Approximation Theory 102, 120–140 (2000).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [H-P] F. Hiai and D. Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and monographs 77, AMS, (2000).

    Google Scholar 

  • [Jo1] K. Johansson, On fluctuations of eigenvalues of random hermitian matrices, Duke Mathematical Journal 91, 151–204 (1998).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [Jo2] K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209, 437–476 (2000).

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [Jo3] K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Annals Comm. Math. 153, 259–296 (2001).

    CrossRef  MATH  Google Scholar 

  • [Jo4] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields 123, 225–280 (2003).

    CrossRef  Google Scholar 

  • [K-V] A.B. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal polynomials with varying reccurrence coefficients, Journal of Approx. Theory 99, 167–197 (1999).

    CrossRef  MATH  Google Scholar 

  • [S-T] E.B. Saff and V. Totik, Logarithmic potentials with external fields, Grundlehren Mathematischen Wissenschaften 316, Springer, (1997).

    Google Scholar 

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Féral, D. (2008). On large deviations for the spectral measure of discrete Coulomb gas. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_2

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