Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles
- 278 Downloads
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.
KeywordsRandom matrices Beta ensembles Entropy
We are grateful to Alexei Borodin and Kurt Johansson for useful discussions, and to Alain Rouault for bringing our attention to the paper  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.
- 6.Borodin, A., Serfaty, S.: Renormalized energy concentration in random matrices. arXiv:1201.2853v2 [math.PR]
- 7.Borot, G., Guionnet, A.: Asymptotic expansion of β-matrix models in the one-cut regime. arXiv:1107.1167
- 8.Bourgade, P., Erdös, L., Yau, H.T.: Universality of general β-ensembles. arXiv:1104.2272 [math.PR]
- 14.Dumitriu, I., Paquette, E.: Global fluctuations for linear statistics of β-Jacobi ensembles. arXiv:1203.6103 [math.PR]
- 17.Erdös, L., Yau, H.T.: Gap universality of generalized Wigner and beta-ensembles. arXiv:1211.3786 [math.PR]
- 26.Mkrtchyan, S.: Entropy of Schur-Weyl measures. Ann. Inst. Henri Poincaré, B, to appear. arXiv:1107.1541v1 [math.RT]
- 32.Shcherbina, M.: Fluctuations of linear eigenvalue statistics of β matrix models in the multi-cut regime. arXiv:1205.7062 [math-ph]
- 33.Sosoe, P., Wong, P.: Local semicircle law in the bulk for Gaussian β-ensemble. arXiv:1112.2016 [math.PR]