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Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles. (English) Zbl 1283.60012
Summary: We show that beta ensembles in the 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.

MSC:
60B20 Random matrices (probabilistic aspects)
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[1] Albeverio, S.; Pastur, L.; Shcherbina, M., On the 1/\(n\) expansion for some unitary invariant ensembles of random matrices, Commun. Math. Phys., 224, 271-305, (2001) · Zbl 1038.82039
[2] Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009) · Zbl 1184.15023
[3] Baker, T.H.; Forrester, P.J., Finite-\(N\) fluctuation formulas for random matrices, J. Stat. Phys., 88, 1371-1386, (1997) · Zbl 0939.82020
[4] Barnes, E.W., The theory of the G-function, Quart. J. Pure Appl. Math., 31, 264-313, (1900) · JFM 30.0389.02
[5] Ben Arous, G.; Guionnet, A., Large deviations for wigner’s law and voiculescu non-commutative entropy, Probab. Theory Relat. Fields, 108, 517-542, (1997) · Zbl 0954.60029
[6] Borodin, A., Serfaty, S.: Renormalized energy concentration in random matrices. arXiv:1201.2853v2 [math.PR] · Zbl 1276.60007
[7] Borot, G., Guionnet, A.: Asymptotic expansion of \(β\)-matrix models in the one-cut regime. arXiv:1107.1167 · Zbl 1344.60012
[8] Bourgade, P., Erdös, L., Yau, H.T.: Universality of general \(β\)-ensembles. arXiv:1104.2272 [math.PR] · Zbl 1239.60005
[9] Bourgade, P.; Erdös, L.; Yau, H.T., Bulk universality of general \(β\)-ensembles with non-convex potential, J. Math. Phys., 53, (2012) · Zbl 1278.82032
[10] Bufetov, A., On the vershik-kerov conjecture concerning the Shannon-mcmillan-breiman theorem for the Plancherel family of measures on the space of Young diagrams, Geom. Funct. Anal., 22, 938-975, (2012) · Zbl 1254.05024
[11] Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Am. Math. Soc., Providence (2000) · Zbl 0997.47033
[12] Dumitriu, I.; Edelman, A., Matrix models for beta ensembles, J. Math. Phys., 43, 5830-5847, (2002) · Zbl 1060.82020
[13] Dumitriu, I.; Edelman, A., Global spectrum fluctuations for the \(β\)-Hermite and \(β\)-Laguerre ensembles via matrix models, J. Math. Phys., 47, (2006) · Zbl 1112.82021
[14] Dumitriu, I., Paquette, E.: Global fluctuations for linear statistics of \(β\)-Jacobi ensembles. arXiv:1203.6103 [math.PR] · Zbl 1268.60009
[15] Dyson, F.J., Statistical theory of the energy levels of complex systems. I, J. Math. Phys., 3, 140-156, (1962) · Zbl 0105.41604
[16] Dyson, F.J., Statistical theory of the energy levels of complex systems. III, J. Math. Phys., 3, 166-175, (1962) · Zbl 0105.41604
[17] Erdös, L., Yau, H.T.: Gap universality of generalized Wigner and beta-ensembles. arXiv:1211.3786 [math.PR]
[18] Forrester, P.J.: Log-gases and Random Matrices. Princeton University Press, Princeton (2010) · Zbl 1217.82003
[19] Forrester, P.J., Large deviation eigenvalue density for the soft edge Laguerre and Jacobi \(β\)-ensembles, J. Phys. A, 45, (2012) · Zbl 1241.82007
[20] Forrester, P.J., Spectral density asymptotics for Gaussian and Laguerre \(β\)-ensembles in the exponentially small region, J. Phys. A, 45, (2012) · Zbl 1237.15032
[21] Johansson, K., On szegö’s asymptotic formula for Toeplitz determinants and generalizations, Bull. Sci. Math., 112, 257-304, (1988) · Zbl 0661.30001
[22] Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91, 151-204, (1998) · Zbl 1039.82504
[23] Killip, R.; Nenciu, I., Matrix models for circular ensembles, Int. Math. Res. Not., 50, 2665-2701, (2004) · Zbl 1255.82004
[24] Mehta, M.L.: Random Matrices. Elsevier/Academic Press, San Diego (2004) · Zbl 1107.15019
[25] Mkrtchyan, S., Asymptotics of the maximal and the typical dimensions of isotypic components of tensor representations of the symmetric group, Eur. J. Comb., 33, 1631-1652, (2012) · Zbl 1248.20012
[26] Mkrtchyan, S.: Entropy of Schur-Weyl measures. Ann. Inst. Henri Poincaré, B, to appear. arXiv:1107.1541v1 [math.RT] · Zbl 1039.82504
[27] Pastur, L., Limiting laws of linear eigenvalue statistics for Hermitian matrix models, J. Math. Phys., 47, (2006) · Zbl 1112.82022
[28] Popescu, I., Talagrand inequality for the semicircular law and energy of the eigenvalues of beta ensembles, Math. Res. Lett., 14, 1023-1032, (2007) · Zbl 1142.46031
[29] Ramírez, J.; Rider, B.; Virág, B., Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Am. Math. Soc., 24, 919-944, (2011) · Zbl 1239.60005
[30] Selberg, A., Remarks on a multiple integral, Norsk Mat. Tidsskr., 26, 71-78, (1944) · Zbl 0063.06870
[31] Shcherbina, M., Orthogonal and symplectic matrix models: universality and other properties, Commun. Math. Phys., 307, 761-790, (2011) · Zbl 1232.15027
[32] Shcherbina, M.: Fluctuations of linear eigenvalue statistics of \(β\) matrix models in the multi-cut regime. arXiv:1205.7062 [math-ph] · Zbl 1273.15042
[33] Sosoe, P., Wong, P.: Local semicircle law in the bulk for Gaussian \(β\)-ensemble. arXiv:1112.2016 [math.PR] · Zbl 1250.82020
[34] Valkó, B.; Virág, B., Continuum limits of random matrices and the Brownian carousel, Invent. Math., 177, 463-508, (2009) · Zbl 1204.60012
[35] Vershik, A.M.; Kerov, S.V., Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funkc. Anal. Prilozh., 19, 25-36, (1985)
[36] Wong, P., Local semicircle law at the spectral edge for Gaussian \(β\)-ensembles, Commun. Math. Phys., 312, 251-263, (2012) · Zbl 1251.82031
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