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Journal of Statistical Physics

, Volume 152, Issue 1, pp 1–14 | Cite as

Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles

  • Alexander Bufetov
  • Sevak Mkrtchyan
  • Maria Shcherbina
  • Alexander Soshnikov
Article

Abstract

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.

Keywords

Random matrices Beta ensembles Entropy 

Notes

Acknowledgements

We are grateful to Alexei Borodin and Kurt Johansson for useful discussions, and to Alain Rouault for bringing our attention to the paper [28] by I. Popescu.

A. Bufetov has been supported in part by an Alfred P. Sloan Research Fellowship, a Dynasty Foundation Fellowship, as well as an IUM-Simons Fellowship, by the Grant MK-6734.2012.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by the RFBR-CNRS grant 10-01-93115-NTsNIL and by the RFBR grant 11-01-00654.

M. Shcherbina has been supported in part by the project “Ukrainian branch of the French-Russian Poncelet laboratory”—“Probability problems on groups and spectral theory”.

A. Soshnikov has been supported in part by the NSF grant DMS-1007558.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander Bufetov
    • 1
    • 2
    • 3
    • 4
    • 5
    • 6
  • Sevak Mkrtchyan
    • 7
  • Maria Shcherbina
    • 8
  • Alexander Soshnikov
    • 9
  1. 1.Laboratoire d’Analyse, Topologie, ProbabilitésCNRSMarseilleFrance
  2. 2.The Steklov Institute of MathematicsMoscowRussia
  3. 3.The Institute for Information Transmission ProblemsMoscowRussia
  4. 4.National Research University Higher School of EconomicsMoscowRussia
  5. 5.The Independent University of MoscowMoscowRussia
  6. 6.Rice UniversityHoustonUSA
  7. 7.Carnegie Mellon UniversityPittsburghUSA
  8. 8.Institute for Low Temperature Physics Ukr. Ac. Sci.KharkovUkraine
  9. 9.University of California at DavisDavisUSA

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