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Edge universality of beta ensembles. (English) Zbl 1306.82010
The first main result of the paper is the edge universality for the beta ensembles for any \(\beta\geq 1\) under the following assumptions: (1) the equilibrium density is supported on a finite interval and (2) the potential is \(\mathcal C^4\) with at least logarithmic growth at infinity, bounded from below by the second derivative, and regular in the sense of A. B. J. Kuijlaars and K. T. R. McLaughlin [Commun. Pure Appl. Math. 53, No. 6, 736–785 (2000; Zbl 1022.31001)] (which means that the equilibrium density is positive in the interior of the interval of support and vanishes like a square root at each of the endpoints of the interval). The proof is based on a certain rigidity estimate for locations of particles. This estimate, in addition, allows to extend existing results on the bulk universality for the beta ensembles from analytic to \(\mathcal C^4\) potentials. (The universality for correlation functions holds for all \(\beta>0\), and for gaps for all \(\beta\geq 1\).) Unlike the main result, the rigidity estimate holds for all \(\beta>0\). The restriction on \(\beta\) in the universality result is due to the use of the local Dyson Brownian motion, which is well defined only for \(\beta\geq 1\). The authors believe that this restriction can be removed, but do not pursue this question in the present paper. Around the same time, the edge universality of the beta ensembles for all \(\beta>0\) and convex polynomial potentials was proved in [M. Krishnapur, B. Rider and B. Virag, “Universality of the stochastic Airy operator”, Preprint, arXiv:1306.4832].
The second main result of the paper is the edge universality of generalized Wigner matrices for all symmetry classes under the assumption of subexponential decay of normalized matrix elements. This extends previous works of various authors on edge universality of Wigner matrices, which essentially relied on the fact that the variances of the matrix elements were identical, to generalized Wigner matrices (with non-identical variances). The edge universality of statistics of generalized Wigner matrices was shown in [L. Erdős et al., Adv. Math. 229, No. 3, 1435–1515 (2012; Zbl 1238.15017)], but it was not proved there that the statistics are independent of the variances. This is achieved in the present work.

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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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. In: Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, vol. 118 (2010) · Zbl 1184.15023
[3] Auffinger, A.; Ben Arous, G.; Péché, S., Poisson convergence for the largest eigenvalues of heavy tailed random matrices, Ann. Inst. Henri Poincaré Probab. Stat., 45, 589-610, (2009) · Zbl 1177.15037
[4] Bakry, D.; Émery, M., Diffusions hypercontractives, Séminaire de Probabilités XIX, 1123, 117-206, (1983) · Zbl 0561.60080
[5] Ben Arous, G.; Dembo, A.; Guionnet, A., Aging of spherical spin glasses, Probab. Theory Related Fields, 120, 1-67, (2001) · Zbl 0993.60055
[6] Ben Arous, G.; Guionnet, A., Large deviations for wigner’s law and voiculescu’s non-commutative entropy, Probab. Theory Related Fields, 108, 517-542, (1997) · Zbl 0954.60029
[7] Bleher, P.; Its, A., Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. Math. (2), 150, 185-266, (1999) · Zbl 0956.42014
[8] Bourgade, P.; Erdös, L.; Yau, H.-T., Universality of general \(β\)-ensembles, Duke Math. J., 163, 1127-1190, (2014) · Zbl 1298.15040
[9] Bourgade, P., Erdös, L., Yau, H.-T.: Bulk universality of general \(β\)-ensembles with non-convex potential. J. Math. Phys. 53 (2012) · Zbl 1238.15017
[10] Boutet de Monvel, A.; Pastur, L.; Shcherbina, M., On the statistical mechanics approach in the random matrix theory, integrated density of states, J. Stat. Phys., 79, 585-611, (1995) · Zbl 1081.82569
[11] Caffarelli, L.; Chan, C.H.; Vasseur, A., Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24, 849-869, (2011) · Zbl 1223.35098
[12] Deift, P.; Gioev, D., Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices, Comm. Pure Appl. Math., 60, 867-910, (2007) · Zbl 1119.15022
[13] Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. IMRP 2, Art. ID rpm004, 116 (2007) · Zbl 1136.82021
[14] Deift, P.; Gioev, D.; Kriecherbauer, T.; Vanlessen, M., Universality for orthogonal and symplectic Laguerre-type ensembles, J. Stat. Phys., 129, 949-1053, (2007) · Zbl 1136.15014
[15] Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math., 52, 1491-1552, (1999) · Zbl 1026.42024
[16] Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math., 52, 1335-1425, (1999) · Zbl 0944.42013
[17] Deuschel, J.-D.; Giacomin, G.; Ioffe, D., Large deviations and concentration properties for \({∇φ}\) interface models, Probab. Theory Related Fields, 117, 49-111, (2000) · Zbl 0988.82018
[18] Dumitriu, I.; Edelman, A., Matrix models for beta ensembles, J. Math. Phys., 43, 5830-5847, (2002) · Zbl 1060.82020
[19] Erdős, L.; Knowles, A.; Yau, H.-T.; Yin, J., Spectral statistics of Erdös-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues, Comm. Math. Phys., 314, 587-640, (2012) · Zbl 1251.05162
[20] Erdős , L.; Péché, S.; Ramírez, J.A.; Schlein, B.; Yau, H.-T., Bulk universality for Wigner matrices, Comm. Pure Appl. Math., 63, 895-925, (2010) · Zbl 1216.15025
[21] Erdős, L.; Ramírez, J.; Schlein, B.; Tao, T.; Vu, V.; Yau, H.-T., Bulk universality for Wigner Hermitian matrices with subexponential decay, Math. Res. Lett., 17, 667-674, (2010) · Zbl 1277.15027
[22] Erdős, L.; Ramírez, J.; Schlein, B.; Yau, H.-T., Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation, Electron. J. Prob., 15, 526-604, (2010) · Zbl 1225.15034
[23] Erdős, L.; Schlein, B.; Yau, H.-T., Universality of random matrices and local relaxation flow, Invent. Math., 185, 75-119, (2011) · Zbl 1225.15033
[24] Erdős, L.; Schlein, B.; Yau, H.-T.; Yin, J., The local relaxation flow approach to universality of the local statistics of random matrices, Ann. Inst. Henri Poincaré (B), 48, 1-46, (2012) · Zbl 1285.82029
[25] Erdős, L., Yau, H.-T.: Gap universality of generalized Wigner and beta ensembles. arXiv:1211.3786 (2012) · Zbl 1060.82020
[26] Erdős L., Yau H.-T. (2012) A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 17, 28, 5 · Zbl 1245.15038
[27] Erdős, L.; Yau, H.-T., Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc. (N.S.), 49, 377-414, (2012) · Zbl 1263.15032
[28] Erdős, L.; Yau, H.-T.; Yin, J., Universality for generalized Wigner matrices with Bernoulli distribution, J. Comb., 2, 15-81, (2011) · Zbl 1235.15029
[29] Erdős, L.; Yau, H.-T.; Yin, J., Bulk universality for generalized Wigner matrices, Probab. Theory Related Fields, 154, 341-407, (2012) · Zbl 1277.15026
[30] Erdős, L.; Yau, H.-T.; Yin, J., Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math., 229, 1435-1515, (2012) · Zbl 1238.15017
[31] Eynard, B.: Master loop equations, free energy and correlations for the chain of matrices. J. High Energy Phys. 11 (2003) · Zbl 1081.82569
[32] Forrester, P.J.: Log-gases and random matrices. In: London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton, NJ (2010) · Zbl 1217.82003
[33] Giacomin, G.; Olla, S.; Spohn, H., Equilibrium fluctuations for \({∇φ}\) interface model, Ann. Probab., 29, 1138-1172, (2001) · Zbl 1017.60100
[34] Gustavsson, J., Gaussian fluctuations of eigenvalues in the GUE, Ann. Inst. H. Poincare Probab. Stat., 41, 151-178, (2005) · Zbl 1073.60020
[35] Helffer, B.; Sjöstrand, J., On the correlation for Kac-like models in the convex case, J. Stat. Phys., 74, 349-409, (1994) · Zbl 0946.35508
[36] Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91, 151-204, (1998) · Zbl 1039.82504
[37] Kipnis, C.; Varadhan, S.R.S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys., 104, 1-19, (1986) · Zbl 0588.60058
[38] Krishnapur, M., Rider, B., Virag, B.: Universality of the Stochastic Airy Operator. Preprint arXiv:1306.4832 (2013) · Zbl 1136.15015
[39] Kuijlaars, A.B.J.; McLaughlin, K.T.-R., Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math., 53, 736-785, (2000) · Zbl 1022.31001
[40] Ledoux, M.; Rider, B., Small deviations for beta ensembles, Electron. J. Probab., 15, 1319-1343, (2010) · Zbl 1228.60015
[41] Lee, J.-O.; Yin, J., A necessary and sufficient condition for edge universality of Wigner matrices, Duke Math. J., 163, 117-173, (2014) · Zbl 1296.60007
[42] Naddaf, A.; Spencer, T., On homogenization and scaling limit of some gradient perturbations of a massless free field, Commun. Math. Phys., 183, 55-84, (1997) · Zbl 0871.35010
[43] O’Rourke, S., Gaussian fluctuations of eigenvalues in Wigner random matrices, J. Stat. Phys., 138, 1045-1066, (2010) · Zbl 1196.15036
[44] Pastur, L.; Shcherbina, M., On the edge universality of the local eigenvalue statistics of matrix models, Mat. Fiz. Anal. Geom., 10, 335-365, (2003) · Zbl 1064.60011
[45] Pastur, L.; Shcherbina, M., Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Stat. Phys., 86, 109-147, (1997) · Zbl 0916.15009
[46] Pastur, L.; Shcherbina, M., Bulk universality and related properties of Hermitian matrix models, J. Stat. Phys., 130, 205-250, (2008) · Zbl 1136.15015
[47] Péché, S.; Soshnikov, A., Wigner random matrices with non-symmetrically distributed entries, J. Stat. Phys., 129, 857-884, (2007) · Zbl 1139.82019
[48] Ramírez, J.A.; Rider, B.; Virág, B., Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc., 24, 919-944, (2011) · Zbl 1239.60005
[49] Shcherbina, M., Edge universality for orthogonal ensembles of random matrices, J. Stat. Phys., 136, 35-50, (2009) · Zbl 1175.15031
[50] Shcherbina, M., Orthogonal and symplectic matrix models: universality and other properties, Comm. Math. Phys., 307, 761-790, (2011) · Zbl 1232.15027
[51] Soshnikov, A., Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys., 207, 697-733, (1999) · Zbl 1062.82502
[52] Stein, E.M.; Weiss, G., Fractional integrals on \(n\)-dimensional Euclidean space, J. Math. Mech., 7, 503-514, (1958) · Zbl 0082.27201
[53] Tao, T., The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, Probab. Theory Related Fields, 157, 81-106, (2013) · Zbl 1280.15023
[54] Tao, T.; Vu, V., Random matrices: universality of local eigenvalue statistics up to the edge, Comm. Math. Phys., 298, 549-572, (2010) · Zbl 1202.15038
[55] Tao, T.; Vu, V., Random matrices: universality of local eigenvalue statistics, Acta Math., 206, 1-78, (2011) · Zbl 1217.15043
[56] Tracy, C.; Widom, H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys., 159, 151-174, (1994) · Zbl 0789.35152
[57] Tracy, C.; Widom, H., On orthogonal and symplectic matrix ensembles, Comm. Math. Phys., 177, 727-754, (1996) · Zbl 0851.60101
[58] Valkó, B.; Virág, B., Continuum limits of random matrices and the Brownian carousel, Invent. Math., 177, 463-508, (2009) · Zbl 1204.60012
[59] Widom, H., On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Stat. Phys., 94, 347-363, (1999) · Zbl 0935.60090
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