zbMATH — the first resource for mathematics

CLT for fluctuations of \(\beta \)-ensembles with general potential. (English) Zbl 1406.60036
Summary: We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or \(\beta \)-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of K. Johansson [Duke Math. J. 91, No. 1, 151–204 (1998; Zbl 1039.82504)], G. Borot and A. Guionnet [Commun. Math. Phys. 317, No. 2, 447–483 (2013; Zbl 1344.60012); “Asymptotic expansion of \(\beta\) matrix models in the multi-cut regime”, Preprint, arXiv:1303.1045] and M. Shcherbina [J. Stat. Phys. 151, No. 6, 1004–1034 (2013; Zbl 1273.15042)]. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

60F05 Central limit and other weak theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B10 Convergence of probability measures
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
82B05 Classical equilibrium statistical mechanics (general)
60G15 Gaussian processes
Full Text: DOI Euclid arXiv
[1] [AKM17] S. Armstrong, T. Kuusi, and J.-C. Mourrat. Quantitative stochastic homogenization and large-scale regularity. https://arxiv.org/abs/1705.05300, 2017.
[2] [BdMPS95] A. Boutet de Monvel, L. Pastur, and M. Shcherbina. On the statistical mechanics approach in the random matrix theory: integrated density of states. Journal of statistical physics, 79(3):585–611, 1995. · Zbl 1081.82569
[3] [BEY14] P. Bourgade, L. Erdős, L.s, and H.-T. Yau. Universality of general \(β \) -ensembles. Duke Math. J., 163(6):1127–1190, 04 2014. · Zbl 1298.15040
[4] [BFG13] F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for \(β \)-matrix models and universality. Communications in Mathematical Physics, 338(2):589–619, 2013. · Zbl 1330.49046
[5] [BG13a] G. Borot and A. Guionnet. Asymptotic expansion of \(β \) matrix models in the multi-cut regime. arXiv preprint arXiv:1303.1045, 2013. · Zbl 1344.60012
[6] [BG13b] G. Borot and A. Guionnet. Asymptotic expansion of \(β \) matrix models in the one-cut regime. Communications in Mathematical Physics, 317(2):447–483, 2013. · Zbl 1344.60012
[7] [BL18] F. Bekerman and A. Lodhia. Mesoscopic central limit theorem for general \(β \)-ensembles. Annales de l’Institut Henri Poincaré (to appear), 2018. · Zbl 1417.60006
[8] [CK06] T. Claeys and A.B.J. Kuijlaars. Universality of the double scaling limit in random matrix models. Communications on Pure and Applied Mathematics, 59(11):1573–1603, 2006. · Zbl 1111.35031
[9] [CKI10] T. Claeys, I. Krasovsky, and A. Its. Higher-order analogues of the Tracy-Widom distribution and the Painlevé ii hierarchy. Communications on pure and applied mathematics, 63(3):362–412, 2010. · Zbl 1198.34191
[10] [Cla08] T. Claeys. Birth of a cut in unitary random matrix ensembles. International Mathematics Research Notices, 2008:rnm166, 2008. · Zbl 1141.82304
[11] [DKM98] P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin. New results on the equilibrium measure for logarithmic potentials in the presence of an external field. Journal of approximation theory, 95(3):388–475, 1998. · Zbl 0918.31001
[12] [DKM\(^{+}\)99] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Communications on Pure and Applied Mathematics, 52(11):1335–1425, 1999. · Zbl 0944.42013
[13] [For10] P. Forrester. Log-gases and random matrices. Princeton University Press, 2010.
[14] [GMS07] A. Guionnet and E. Maurel-Segala. Second order asymptotics for matrix models. The Annals of Probability, 35(6):2160–2212, 2007. · Zbl 1129.15020
[15] [GS14] A. Guionnet and D. Shlyakhtenko. Free monotone transport. Inventiones mathematicae, 197(3):613–661, 2014. · Zbl 1312.46059
[16] [Joh98] K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices. Duke Mathematical Journal, 91(1):151–204, 1998. · Zbl 1039.82504
[17] [KM00] A. Kuijlaars and K. McLaughlin. Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. Communications on Pure and Applied Mathematics, 53(6):736–785, 2000. · Zbl 1022.31001
[18] [LLW17] G. Lambert, M. Ledoux, and C. Webb. Stein’s method for normal approximation of linear statistics of beta-ensembles. https://arxiv.org/abs/1706.10251, 06 2017.
[19] [LS17] T. Leblé and S. Serfaty. Large deviation principle for empirical fields of Log and Riesz gases. Inventiones mathematicae, 210(3):645–757, 2017. · Zbl 1397.82007
[20] [LS18] T. Leblé and S. Serfaty. Fluctuations of two dimensional Coulomb gases. Geometric and Functional Analysis, 28(2):443–508, 2018. · Zbl 1423.60045
[21] [MdMS14] M. Maï da and É. Maurel-Segala. Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices. Probab. Theory Related Fields, 159(1-2):329–356, 2014.
[22] [Mo08] M.Y. Mo. The Riemann–Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. International Mathematics Research Notices, 2008.
[23] [Mus92] N. I. Muskhelishvili. Singular integral equations. Dover Publications, Inc., New York, 1992.
[24] [PS14] M. Petrache and S. Serfaty. Next order asymptotics and renormalized energy for riesz interactions. Journal of the Institute of Mathematics of Jussieu, pages 1–69, 2014.
[25] [Shc13] M. Shcherbina. Fluctuations of linear eigenvalue statistics of \(β \) matrix models in the multi-cut regime. Journal of Statistical Physics, 151(6):1004–1034, 2013. · Zbl 1273.15042
[26] [Shc14] M. Shcherbina. Change of variables as a method to study general \(β \)-models: bulk universality. Journal of Mathematical Physics, 55(4):043504, 2014.
[27] [SS15] É. Sandier and S. Serfaty. 1D log gases and the renormalized energy: crystallization at vanishing temperature. Probability Theory and Related Fields, 162(3-4):795–846, 2015.
[28] [ST13] E.B. Saff and V. Totik. Logarithmic potentials with external fields, volume 316. Springer Science & Business Media, 2013.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.