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From Gumbel to Tracy-Widom
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  • Published: 30 June 2006

From Gumbel to Tracy-Widom

  • K. Johansson1 

Probability Theory and Related Fields volume 138, pages 75–112 (2007)Cite this article

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Abstract

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( − exp( − x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp( − x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, \(M=M_0+\sqrt{2S} V\), with M 0 diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.

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References

  1. Adler M., van Moerbeke P. (2005). PDEs for the joint distributions of the Dyson, airy and sine processes. Ann. Probab. 33: 1326–1361

    Article  MATH  MathSciNet  Google Scholar 

  2. Daley D.J., Vere-Jones D. (2003). An Introduction to the Theory of Point Processes, vol. 1, 2nd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  3. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52: 1491–1552

    Article  MATH  MathSciNet  Google Scholar 

  4. Forrester P.J. (1993). Statistical properties of the eigenvalue motion of Hermitian matrices. Phys. Lett. A 173: 355–359

    Article  MathSciNet  Google Scholar 

  5. Forrester P.J., Nagao T. (1998). Correlations for the circular Dyson Brownian motion model with Poisson initial conditions. Nuclear Phys. B 532: 733–752

    Article  MATH  MathSciNet  Google Scholar 

  6. Frahm K.M., Guhr T., Müller-Groeling A. (1998). Between Poisson and GUE statistics: role of the Breit Wigner width. Ann. Phys. 270: 292–327

    Article  MATH  Google Scholar 

  7. Garcia-Garcia A.M., Verbaarschot J.J.M. (2000). Chiral random matrix model for critical statistics. Nucl.Phys. B 586: 668–685

    Article  MATH  MathSciNet  Google Scholar 

  8. Gohberg I., Goldberg S., Krupnik N. (2000). Traces and Determinants of Linear Operators. Birkhäuser, Basel

    MATH  Google Scholar 

  9. Gravner J., Tracy C.A., Widom H. (2002). A growth model in a random environment. Ann. Probab. 30: 1340–1368

    Article  MATH  MathSciNet  Google Scholar 

  10. Guhr T. (1996). Transitions toward quantum chaos: with supersymmetry from Poisson to Gauss. Ann. Phys. 250: 145–192

    Article  MATH  MathSciNet  Google Scholar 

  11. Johansson K. (2000). Shape fluctuations and random matrices. Commun. Math. Phys. 209: 437–476

    Article  MATH  MathSciNet  Google Scholar 

  12. Johansson K. (2001). Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215: 683–705

    Article  MATH  MathSciNet  Google Scholar 

  13. Johansson K. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123: 225–280

    Article  MATH  MathSciNet  Google Scholar 

  14. Johansson, K.: Toeplitz determinants, random growth and determinantal processes. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 53–62. Higher Ed. Press, Beijing (2002)

  15. Johansson K. (2003). Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242: 277–329

    MATH  MathSciNet  Google Scholar 

  16. Johansson, K.: Random Matrices and determinantal processes. Lecture notes from the Les Houches summer school on Mathematical Statistical Physics (2005). arXiv:math-ph/0510038

  17. Karlin S., McGregor G. (1959). Coincidence probabilities. Pacific J. Math 9: 1141–1164

    MATH  MathSciNet  Google Scholar 

  18. Krasikov I. (2004). New bounds on the Hermite polynomials. East J. Approx. 10: 355–362

    MATH  MathSciNet  Google Scholar 

  19. Khorunzhy A., Hirsch W. (2002). On asymptotic expansions and scales of spectral universality in band matrix ensembles. Commun. Math. Phys. 231: 223–255

    Article  MATH  Google Scholar 

  20. Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics, Springer, Berlin Heidelberg New York (1983)

  21. Moshe M., Neuberger H., Shapiro B. (1994). Generalized ensemble of random matrices. Phys. Rev. Lett. 73: 1497–1500

    Article  MATH  MathSciNet  Google Scholar 

  22. Muttalib K.A., Chen Y., Ismail M.E.H., Nicopoulos V.N. (1993). New family of unitary random matrices. Phys. Rev. Lett. 71: 471–475

    Article  MATH  MathSciNet  Google Scholar 

  23. Okounkov A. (2002). Generating functions for intersection numbers on moduli spaces of curves. Int. Math. Res. Not. 18: 933–957

    Article  MathSciNet  Google Scholar 

  24. Pandey A. (1995). Brownian-motion model of discrete spectra. Chaos Solitons Fractals 5: 1275—1285

    Article  MATH  MathSciNet  Google Scholar 

  25. Ruzmaikina A., Aizenman M. (2005). Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33: 82–113

    Article  MATH  MathSciNet  Google Scholar 

  26. Simon, B.: Trace ideals and their applications, 2nd edn. Mathematical Surveys and Monographs, vol. 120. American Mathematical Society, Providence (2005)

  27. Soshnikov A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207: 697–733

    Article  MATH  MathSciNet  Google Scholar 

  28. Soshnikov A. (2000). Determinantal random point fields. Russian Math. Surv. 55: 923–975

    Article  MATH  MathSciNet  Google Scholar 

  29. Soshnikov A. (2004). Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9: 82–91, (electronic)

    MATH  MathSciNet  Google Scholar 

  30. Tracy C.A., Widom H. (1994). Level spacing distributions and the airy kernel. Commun. Math. Phys. 159: 151–174

    Article  MATH  MathSciNet  Google Scholar 

  31. Tracy, C.A., Widom, H.: Distribution functions for largest eigenvalues and their applications. In: Proceedings of the International Congress of Mathematicians, vol. I, pp. 587–596, Higher Ed. Press, Beijing (2002)

  32. Vershik, A., Yakubovich, Yu.: Fluctuation of the maximal particle energy of the quantum ideal gas and random partitions. arXiv:math-ph/0501043

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Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    K. Johansson

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  1. K. Johansson
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Correspondence to K. Johansson.

Additional information

Supported by the Göran Gustafsson Foundation (KVA).

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Johansson, K. From Gumbel to Tracy-Widom. Probab. Theory Relat. Fields 138, 75–112 (2007). https://doi.org/10.1007/s00440-006-0012-7

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  • Received: 10 October 2005

  • Revised: 05 April 2006

  • Published: 30 June 2006

  • Issue Date: May 2007

  • DOI: https://doi.org/10.1007/s00440-006-0012-7

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Keywords

  • Point Process
  • Random Matrix
  • Determinantal Process
  • Trace Class Operator
  • Random Matrix Model
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