Electronic Journal of Probability

CLT for Fluctuations of $\beta $-ensembles with general potential

Florent Bekerman, Thomas Leblé, and Sylvia Serfaty

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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 Johansson, Borot-Guionnet and Shcherbina. 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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 115, 31 pp.

Received: 6 February 2018
Accepted: 4 August 2018
First available in Project Euclid: 24 November 2018

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Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B05: Classical equilibrium statistical mechanics (general) 60G15: Gaussian processes

beta-ensembles log gas central limit theorem linear statistics

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Bekerman, Florent; Leblé, Thomas; Serfaty, Sylvia. CLT for Fluctuations of $\beta $-ensembles with general potential. Electron. J. Probab. 23 (2018), paper no. 115, 31 pp. doi:10.1214/18-EJP209. https://projecteuclid.org/euclid.ejp/1543028703

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  • [AKM17] S. Armstrong, T. Kuusi, and J.-C. Mourrat. Quantitative stochastic homogenization and large-scale regularity. https://arxiv.org/abs/1705.05300, 2017.
  • [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.
  • [BEY14] P. Bourgade, L. Erdős, L.s, and H.-T. Yau. Universality of general $\beta $ -ensembles. Duke Math. J., 163(6):1127–1190, 04 2014.
  • [BFG13] F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for $\beta $-matrix models and universality. Communications in Mathematical Physics, 338(2):589–619, 2013.
  • [BG13a] G. Borot and A. Guionnet. Asymptotic expansion of $\beta $ matrix models in the multi-cut regime. arXiv preprint arXiv:1303.1045, 2013.
  • [BG13b] G. Borot and A. Guionnet. Asymptotic expansion of $\beta $ matrix models in the one-cut regime. Communications in Mathematical Physics, 317(2):447–483, 2013.
  • [BL18] F. Bekerman and A. Lodhia. Mesoscopic central limit theorem for general $\beta $-ensembles. Annales de l’Institut Henri Poincaré (to appear), 2018.
  • [CK06] T. Claeys and A.B.J. Kuijlaars. Universality of the double scaling limit in random matrix models. Communications on Pure and Applied Mathematics, 59(11):1573–1603, 2006.
  • [CKI10] T. Claeys, I. Krasovsky, and A. Its. Higher-order analogues of the Tracy-Widom distribution and the Painlevé ii hierarchy. Communications on pure and applied mathematics, 63(3):362–412, 2010.
  • [Cla08] T. Claeys. Birth of a cut in unitary random matrix ensembles. International Mathematics Research Notices, 2008:rnm166, 2008.
  • [DKM98] P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin. New results on the equilibrium measure for logarithmic potentials in the presence of an external field. Journal of approximation theory, 95(3):388–475, 1998.
  • [DKM$^{+}$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.
  • [For10] P. Forrester. Log-gases and random matrices. Princeton University Press, 2010.
  • [GMS07] A. Guionnet and E. Maurel-Segala. Second order asymptotics for matrix models. The Annals of Probability, 35(6):2160–2212, 2007.
  • [GS14] A. Guionnet and D. Shlyakhtenko. Free monotone transport. Inventiones mathematicae, 197(3):613–661, 2014.
  • [Joh98] K. Johansson. On fluctuations of eigenvalues of random Hermitian matrices. Duke Mathematical Journal, 91(1):151–204, 1998.
  • [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.
  • [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.
  • [LS17] T. Leblé and S. Serfaty. Large deviation principle for empirical fields of Log and Riesz gases. Inventiones mathematicae, 210(3):645–757, 2017.
  • [LS18] T. Leblé and S. Serfaty. Fluctuations of two dimensional Coulomb gases. Geometric and Functional Analysis, 28(2):443–508, 2018.
  • [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.
  • [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.
  • [Mus92] N. I. Muskhelishvili. Singular integral equations. Dover Publications, Inc., New York, 1992.
  • [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.
  • [Shc13] M. Shcherbina. Fluctuations of linear eigenvalue statistics of $\beta $ matrix models in the multi-cut regime. Journal of Statistical Physics, 151(6):1004–1034, 2013.
  • [Shc14] M. Shcherbina. Change of variables as a method to study general $\beta $-models: bulk universality. Journal of Mathematical Physics, 55(4):043504, 2014.
  • [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.
  • [ST13] E.B. Saff and V. Totik. Logarithmic potentials with external fields, volume 316. Springer Science & Business Media, 2013.