From sine kernel to Poisson statistics

Romain Allez (Weierstrass Institute)
Laure Dumaz (University of Cambridge)

Abstract


We study the Sine beta process introduced in Valko and Virag, when the inverse temperature beta tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of beta-ensembles and its law is characterised in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine-beta point process converges weakly to a Poisson point process on the real line. Thus, the Sine-beta point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\beta=\infty$) and the Poisson process.

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Pages: 1-25

Publication Date: December 12, 2014

DOI: 10.1214/EJP.v19-3742

References

  • The Oxford handbook of random matrix theory. Edited by Gernot Akemann, Jinho Baik and Philippe Di Francesco. Oxford University Press, Oxford, 2011. xxxii+919 pp. ISBN: 978-0-19-957400-1 MR2920518
  • R. Allez, J.-P. Bouchaud and A. Guionnet. Invariant β-ensembles and the Gauss-Wigner crossover. Phys. Rev. Lett. 109, 094102 (2012).
  • Allez, Romain; Bouchaud, Jean-Philippe; Majumdar, Satya N.; Vivo, Pierpaolo. Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marčenko-Pastur law. J. Phys. A 46 (2013), no. 1, 015001, 22 pp. MR3001575
  • Allez, Romain; Dumaz, Laure. Tracy-Widom at high temperature. J. Stat. Phys. 156 (2014), no. 6, 1146--1183. MR3240875
  • Allez, Romain; Guionnet, Alice. A diffusive matrix model for invariant $\beta$-ensembles. Electron. J. Probab. 18 (2013), no. 62, 30 pp. MR3078021
  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Bai, Zhidong; Silverstein, Jack W. Spectral analysis of large dimensional random matrices. Second edition. Springer Series in Statistics. Springer, New York, 2010. xvi+551 pp. ISBN: 978-1-4419-0660-1 MR2567175
  • Bowick, Mark J.; Brézin, Édouard. Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B 268 (1991), no. 1, 21--28. MR1134369
  • Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Vol. II. General theory and structure. Second edition. Probability and its Applications (New York). Springer, New York, 2008. xviii+573 pp. ISBN: 978-0-387-21337-8 MR2371524
  • Dumaz, Laure; Virag, Balint. The right tail exponent of the Tracy-Widom $\beta$ distribution. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 4, 915--933. MR3127907
  • Dumitriu, Ioana; Edelman, Alan. Matrix models for beta ensembles. J. Math. Phys. 43 (2002), no. 11, 5830--5847. MR1936554
  • Edelman, Alan; Sutton, Brian D. From random matrices to stochastic operators. J. Stat. Phys. 127 (2007), no. 6, 1121--1165. MR2331033
  • Forrester, P. J. Log-gases and random matrices. London Mathematical Society Monographs Series, 34. Princeton University Press, Princeton, NJ, 2010. xiv+791 pp. ISBN: 978-0-691-12829-0 MR2641363
  • Forrester, Peter J. Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region. J. Phys. A 45 (2012), no. 7, 075206, 17 pp. MR2881075
  • Kallenberg, Olav. Random measures. Fourth edition. Akademie-Verlag, Berlin; Academic Press, Inc., London, 1986. 187 pp. ISBN: 0-12-394960-2 MR0854102
  • Killip, Rowan; Stoiciu, Mihai. Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146 (2009), no. 3, 361--399. MR2484278
  • Kritchevski, Eugene; Valko, Benedek; Virag, Balint. The scaling limit of the critical one-dimensional random Schrodinger operator. Comm. Math. Phys. 314 (2012), no. 3, 775--806. MR2964774
  • Mehta, Madan Lal. Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp. ISBN: 0-12-088409-7 MR2129906
  • Nakano, Fumihiko. Level statistics for one-dimensional Schrodinger operators and Gaussian beta ensemble. J. Stat. Phys. 156 (2014), no. 1, 66--93. MR3215116
  • Ramirez, Jose A.; Rider, Brian; Virag, Balint. Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc. 24 (2011), no. 4, 919--944. MR2813333
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Valko, Benedek; Virag, Balint. Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177 (2009), no. 3, 463--508. MR2534097
  • E.P. Wigner. On the statistical distribution of the widths and spacings of nuclear resonance levels, Math. Proc. Cambridge Philos. Soc., 47, 790-798 xiii, 3 (1951).
  • J. Wishart. Generalized product moment distribution in samples. Biometrika 20 A 425 (1928).


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