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On Mesoscopic Equilibrium for Linear Statistics in Dyson’s Brownian Motion
About this Title
Maurice Duits, Department of Mathematics, Royal Institute of Technology (KTH), Lindstedtsvägen 25, SE-10044 Stockholm, Sweden. and Kurt Johansson, Department of Mathematics, Royal Institute of Technology (KTH), Lindstedtsvägen 25, SE-10044 Stockholm, Sweden.
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 255, Number 1222
ISBNs: 978-1-4704-2964-5 (print); 978-1-4704-4821-9 (online)
DOI: https://doi.org/10.1090/memo/1222
Published electronically: August 20, 2018
Keywords: Dyson’s Brownian motion,
Linear Statistics,
Central Limit Theorems,
Mesoscopic scale,
Random Matrices,
Poisson initial conditions
MSC: Primary 60B20, 60F25, 80C44
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of results
- 3. Proof of Theorem
- 4. Proof of Theorem
- 5. Asymptotic analysis of $K_n$ and $R_n$
- 6. Proof of Proposition
- 7. Proof of Lemma
- 8. Random initial points
- 9. Proof of Theorem : the general case
- Appendix
Abstract
In this paper we study mesoscopic fluctuations for Dyson’s Brownian motion with $\beta =2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. In this paper we investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that we consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but we have not yet reached equilibrium at the macrosopic scale. We describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. We consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, we obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.- Yacin Ameur, Håkan Hedenmalm, and Nikolai Makarov, Fluctuations of eigenvalues of random normal matrices, Duke Math. J. 159 (2011), no. 1, 31–81. MR 2817648, DOI 10.1215/00127094-1384782
- Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR 2760897
- Martin Bender, Global fluctuations in general $\beta$ Dyson’s Brownian motion, Stochastic Process. Appl. 118 (2008), no. 6, 1022–1042. MR 2418256, DOI 10.1016/j.spa.2007.07.010
- Alexei Borodin, Biorthogonal ensembles, Nuclear Phys. B 536 (1999), no. 3, 704–732. MR 1663328, DOI 10.1016/S0550-3213(98)00642-7
- Alexei Borodin, Determinantal point processes, The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, pp. 231–249. MR 2932631
- Thierry Cabanal-Duvillard, Fluctuations de la loi empirique de grandes matrices aléatoires, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 3, 373–402 (French, with English and French summaries). MR 1831988, DOI 10.1016/S0246-0203(00)01071-2
- A. Boutet de Monvel and A. Khorunzhy, Asymptotic distribution of smoothed eigenvalue density. I. Gaussian random matrices, Random Oper. Stochastic Equations 7 (1999), no. 1, 1–22. MR 1678012, DOI 10.1515/rose.1999.7.1.1
- A. Boutet de Monvel and A. Khorunzhy, Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices, Random Oper. Stochastic Equations 7 (1999), no. 2, 149–168. MR 1689027, DOI 10.1515/rose.1999.7.2.149
- Jonathan Breuer and Maurice Duits, The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles, Adv. Math. 265 (2014), 441–484. MR 3255467, DOI 10.1016/j.aim.2014.07.026
- Jonathan Breuer and Maurice Duits, Universality of mesoscopic fluctuations for orthogonal polynomial ensembles, Comm. Math. Phys. 342 (2016), no. 2, 491–531. MR 3459158, DOI 10.1007/s00220-015-2514-6
- Jonathan Breuer and Maurice Duits, Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, J. Amer. Math. Soc. 30 (2017), no. 1, 27–66. MR 3556288, DOI 10.1090/S0894-0347-2016-00854-8
- Freeman J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191–1198. MR 148397, DOI 10.1063/1.1703862
- László Erdős and Antti Knowles, The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case, Comm. Math. Phys. 333 (2015), no. 3, 1365–1416. MR 3302637, DOI 10.1007/s00220-014-2119-5
- László Erdős and Antti Knowles, The Altshuler-Shklovskii formulas for random band matrices II: The general case, Ann. Henri Poincaré 16 (2015), no. 3, 709–799. MR 3311888, DOI 10.1007/s00023-014-0333-5
- László Erdős and Horng-Tzer Yau, Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 3, 377–414. MR 2917064, DOI 10.1090/S0273-0979-2012-01372-1
- László Erdős, Horng-Tzer Yau, and Jun Yin, Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math. 229 (2012), no. 3, 1435–1515. MR 2871147, DOI 10.1016/j.aim.2011.12.010
- P. J. Forrester and T. Nagao, Correlations for the circular Dyson Brownian motion model with Poisson initial conditions, Nuclear Phys. B 532 (1998), no. 3, 733–752. MR 1657046, DOI 10.1016/S0550-3213(98)00551-3
- Y. V. Fyodorov, B. A. Khoruzhenko, and N. J. Simm, Fractional Brownian motion with Hurst index $H=0$ and the Gaussian unitary ensemble, Ann. Probab. 44 (2016), no. 4, 2980–3031. MR 3531684, DOI 10.1214/15-AOP1039
- P.J. Forrester and T. Nagao, Multilevel dynamical correlation functions for Dyon’s Brownian motion model of random matrics, Phys. Lett. A 247 (1998), 42–46.
- P. J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010. MR 2641363
- Thomas Guhr and Axel Müller-Groeling, Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales, J. Math. Phys. 38 (1997), no. 4, 1870–1887. Quantum problems in condensed matter physics. MR 1450904, DOI 10.1063/1.531918
- Thomas Guhr, Transitions toward quantum chaos: with supersymmetry from Poisson to Gauss, Ann. Physics 250 (1996), no. 1, 145–192. MR 1407173, DOI 10.1006/aphy.1996.0091
- Yukun He and Antti Knowles, Mesoscopic eigenvalue statistics of Wigner matrices, Ann. Appl. Probab. 27 (2017), no. 3, 1510–1550. MR 3678478, DOI 10.1214/16-AAP1237
- J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág, Determinantal processes and independence, Probab. Surv. 3 (2006), 206–229. MR 2216966, DOI 10.1214/154957806000000078
- Stefan Israelsson, Asymptotic fluctuations of a particle system with singular interaction, Stochastic Process. Appl. 93 (2001), no. 1, 25–56. MR 1819483, DOI 10.1016/S0304-4149(00)00100-9
- Kurt Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), no. 1, 151–204. MR 1487983, DOI 10.1215/S0012-7094-98-09108-6
- Kurt Johansson, Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Comm. Math. Phys. 215 (2001), no. 3, 683–705. MR 1810949, DOI 10.1007/s002200000328
- Kurt Johansson, Random matrices and determinantal processes, Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 1–55. MR 2581882, DOI 10.1016/S0924-8099(06)80038-7
- Kurt Johansson, Universality for certain Hermitian Wigner matrices under weak moment conditions, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 1, 47–79 (English, with English and French summaries). MR 2919198, DOI 10.1214/11-AIHP429
- Wolfgang König, Orthogonal polynomial ensembles in probability theory, Probab. Surv. 2 (2005), 385–447. MR 2203677, DOI 10.1214/154957805100000177
- G. Lambert, Mesoscopic fluctuations for unitary invariant ensembles, (arXiv:1510.03641)
- A. Lodhia and N.J. Simm, Mesoscopic linear statistics of Wigner matrices, (arXiv:1503.03533)
- Russell Lyons, Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 167–212. MR 2031202, DOI 10.1007/s10240-003-0016-0
- Colin McDiarmid, Concentration, Probabilistic methods for algorithmic discrete mathematics, Algorithms Combin., vol. 16, Springer, Berlin, 1998, pp. 195–248. MR 1678578, DOI 10.1007/978-3-662-12788-9_{6}
- A. Soshnikov, Determinantal random point fields, Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 5, 923–975. MR 1799012, DOI 10.1070/rm2000v055n05ABEH000321
- Alexander Soshnikov, The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), no. 3, 1353–1370. MR 1797877, DOI 10.1214/aop/1019160338