Electronic Journal of Probability

The random matrix hard edge: rare events and a transition

Diane Holcomb

Full-text: Open access

Abstract

We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval $[0,\lambda ]$ as $\lambda \to \infty $. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the $\mathrm{Sine} _\beta $ process.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 85, 20 pp.

Dates
Received: 20 November 2017
Accepted: 10 August 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717744

Digital Object Identifier
doi:10.1214/18-EJP212

Mathematical Reviews number (MathSciNet)
MR3858913

Zentralblatt MATH identifier
06964779

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
random matrices point process diffusion Bessel process large deviations central limit theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Holcomb, Diane. The random matrix hard edge: rare events and a transition. Electron. J. Probab. 23 (2018), paper no. 85, 20 pp. doi:10.1214/18-EJP212. https://projecteuclid.org/euclid.ejp/1536717744


Export citation

References

  • [1] Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992, Reprint of the 1972 edition.
  • [2] Folkmar Bornemann and Michael La Croix, The singular values of the GOE, Random Matrices Theory Appl. 4 (2015), no. 2, 1550009, 32.
  • [3] Paul Bourgade, László Erdös, and Horng-Tzer Yau, Edge universality of beta ensembles, Comm. Math. Phys. 332 (2014), no. 1, 261–353.
  • [4] Percy A. Deift, Thomas Trogdon, and Govind Menon, On the condition number of the critically-scaled Laguerre unitary ensemble, Discrete Contin. Dyn. Syst. 36 (2016), no. 8, 4287–4347.
  • [5] Alan Edelman and Michael La Croix, The singular values of the GUE (less is more), Random Matrices Theory Appl. 4 (2015), no. 4, 1550021, 37.
  • [6] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence.
  • [7] Diane Holcomb and Gregorio R. Moreno Flores, Edge scaling of the $\beta $-Jacobi ensemble, J. Stat. Phys. 149 (2012), no. 6, 1136–1160.
  • [8] Diane Holcomb and Benedek Valkó, Large deviations for the ${\rm Sine}_\beta $ and ${\rm Sch}_\tau $ processes, Probab. Theory Related Fields 163 (2015), no. 1-2, 339–378.
  • [9] Diane Holcomb and Benedek Valkó, Overcrowding asymptotics for the $\rm sine_\beta $ process, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 3, 1181–1195.
  • [10] Stéphanie Jacquot and Benedek Valkó, Bulk scaling limit of the Laguerre ensemble, Electron. J. Probab. 16 (2011), no. 11, 314–346.
  • [11] Manjunath Krishnapur, Brian Rider, and Bálint Virág, Universality of the stochastic Airy operator, Comm. Pure Appl. Math. 69 (2016), no. 1, 145–199.
  • [12] Eugene Kritchevski, Benedek Valkó, and Bálint Virág, The scaling limit of the critical one-dimensional random Schrödinger operator, Comm. Math. Phys. 314 (2012), no. 3, 775–806.
  • [13] Thomas G. Kurtz and Philip Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (1991), no. 3, 1035–1070.
  • [14] José A. Ramírez and Brian Rider, Erratum to: Diffusion at the random matrix hard edge [mr2504858], Comm. Math. Phys. 307 (2011), no. 2, 561–563.
  • [15] José A. Ramírez, Brian Rider, and Ofer Zeitouni, Hard edge tail asymptotics, Electron. Commun. Probab. 16 (2011), 741–752.
  • [16] José A. Ramírez and Brian Rider, Diffusion at the random matrix hard edge, Comm. Math. Phys. 288 (2009), no. 3, 887–906.
  • [17] José A. Ramírez, Brian Rider, and Bálint Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc. 24 (2011), no. 4, 919–944.
  • [18] Brian Rider and Patrick Waters, Universality of the stochastic Bessel operator, ArXiv (2016), arXiv:1610.01637.
  • [19] Benedek Valkó and Bálint Virág, Continuum limits of random matrices and the Brownian carousel, Invent. Math. 177 (2009), no. 3, 463–508.