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

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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.

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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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Authors and Affiliations

  1. Laboratoire d’Analyse, Topologie, Probabilités, CNRS, Marseille, France

    Alexander Bufetov

  2. The Steklov Institute of Mathematics, Moscow, Russia

    Alexander Bufetov

  3. The Institute for Information Transmission Problems, Moscow, Russia

    Alexander Bufetov

  4. National Research University Higher School of Economics, Moscow, Russia

    Alexander Bufetov

  5. The Independent University of Moscow, Moscow, Russia

    Alexander Bufetov

  6. Rice University, Houston, USA

    Alexander Bufetov

  7. Carnegie Mellon University, Pittsburgh, USA

    Sevak Mkrtchyan

  8. Institute for Low Temperature Physics Ukr. Ac. Sci., Kharkov, Ukraine

    Maria Shcherbina

  9. University of California at Davis, Davis, USA

    Alexander Soshnikov

Authors
  1. Alexander Bufetov
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  2. Sevak Mkrtchyan
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  3. Maria Shcherbina
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  4. Alexander Soshnikov
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Correspondence to Alexander Soshnikov.

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Bufetov, A., Mkrtchyan, S., Shcherbina, M. et al. Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles. J Stat Phys 152, 1–14 (2013). https://doi.org/10.1007/s10955-013-0761-5

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  • DOI: https://doi.org/10.1007/s10955-013-0761-5

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