Communications in Mathematical Physics

, Volume 359, Issue 1, pp 223–263 | Cite as

Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics

  • Thomas Bothner
  • Robert Buckingham


We analyze the left-tail asymptotics of deformed Tracy–Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability \({1-\gamma\in[0,1]}\). As \({\gamma}\) varies, a transition from Tracy–Widom statistics (\({\gamma=1}\)) to classical Weibull statistics (\({\gamma=0}\)) was observed in the physics literature by Bohigas et al. (Phys Rev E 79:031117, 2009). We provide a description of this transition by rigorously computing the leading-order left-tail asymptotics of the thinned GOE, GUE, and GSE Tracy–Widom distributions. In this paper, we obtain the asymptotic behavior in the non-oscillatory region with \({\gamma\in[0,1)}\) fixed (for the GOE, GUE, and GSE distributions) and \({\gamma\uparrow 1}\) at a controlled rate (for the GUE distribution). This is the first step in an ongoing program to completely describe the transition between Tracy–Widom and Weibull statistics. As a corollary to our results, we obtain a new total-integral formula involving the Ablowitz–Segur solution to the second Painlevé equation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ablowitz M., Segur H.: Asymptotic solutions of the Korteweg–de Vries equation. Stud. Appl. Math. 57, 13–44 (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ablowitz M., Segur H.: Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Physica D 3, 165–184 (1981)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Baik J., Buckingham R., DiFranco J.: Asymptotics of Tracy–Widom distributions and the total integral of a Painlevé II function. Commun. Math. Phys. 280, 463–497 (2008)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Baik J., Buckingham R., DiFranco J., Its A.: Total integrals of Painlevé II solutions. Nonlinearity 22, 1021–1061 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baik J., Deift P., Johansson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12, 1119–1178 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baik J., Deift P., Johansson K.: On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geom. Funct. Anal. 10, 702–731 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baik,J., Rains, E.: Symmetrized random permutations. In: Bleher, P., Its, A. (eds.) Random Matrix Models and Their Applications. MSRI Volume 40 1–19 (2001)Google Scholar
  8. 8.
    Basor E., Widom H.: Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols. J. Funct. Anal. 50, 387–413 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berggren T., Duits M.: Mesoscopic fluctuations for the thinned Circular Unitary Ensemble. Math. Phys. Anal. Geom. 20, 20:19 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bleher P., Kuijlaars A.: Large n limit of Gaussian random matrices with external source, part III: double scaling limit. Commun. Math. Phys. 270, 481–517 (2007)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Bogatskiy A., Claeys T., Its A.: Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge. Commun. Math. Phys. 347, 127–162 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bohigas O., de Carvalho J., Pato M.: Deformations of the Tracy–Widom distribution. Phys. Rev. E 79, 031117 (2009)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bohigas O., Pato M.: Missing levels in correlated spectra. Phys. Lett. B 595, 171–176 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Bohigas O., Pato M.: Randomly incomplete spectra and intermediate statistics. Phys. Rev. E 74, 036212 (2006)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bornemann F., Forrester P., Mays A.: Finite size effects for spacing distributions in random matrix theory: circular ensembles and Riemann zeros. Stud. Appl. Math. 138, 401–437 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Borodin A., Okounkov A., Olshanski G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13, 481–515 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Borot G., Nadal C.: Right tail asymptotic expansion of Tracy–Widom beta laws. Random Matrices Theory Appl. 1, 1250006 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bothner T.: Transition asymptotics for the Painlevé II transcendent. Duke Math. J. 166, 205–324 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bothner T.: From gap probabilities in random matrix theory to eigenvalue expansions. J. Phys. A Math. Theor. 49, 075204 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bothner, T., Buckingham, R.: Large deformations of the Tracy–Widom distribution II. Oscillatory asymptotics (in preparation)Google Scholar
  21. 21.
    Bothner T., Deift P., Its A., Krasovsky I.: On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I. Commun. Math. Phys. 337, 1397–1463 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bothner T., Deift P., Its A., Krasovsky I.: On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II. Oper. Theory Adv. Appl. 259, 213–234 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bothner T., Its A.: Asymptotics of a cubic sine kernel determinant. St. Petersb. Math. J. 26, 22–92 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Budylin A., Buslaev V.: Quasiclassical asymptotics of the resolvent of an integral convolution operator with a sine kernel on a finite interval. Algebra Anal. 7, 79–103 (1995)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Claeys T., Its A., Krasovsky I.: Higher-order analogues of the Tracy–Widom distribution and the Painlevé II hierarchy. Commun. Pure. Appl. Math. 63, 362–412 (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Charlier C., Claeys T.: Thinning and conditioning of the Circular Unitary Ensemble. Random Matrics Theory Appl. 6, 1750007 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Deift P., Its A., Krasovsky I.: Asymptotics of the Airy-kernel determinant. Commun. Math. Phys. 278, 643–678 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Deift P., Its A., Krasovsky I., Zhou X.: The Widom–Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202, 26–47 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Deift P., Zhou X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Dieng M.: Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations. Int. Math. Res. Not. 37, 2263–2287 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dyson F.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ehrhardt T.: Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. Commun. Math. Phys. 262, 317–341 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ehrhardt T.: Dyson’s constants in the asymptotics of the determinants of Wiener–Hopf–Hankel operators with the sine kernel. Commun. Math. Phys. 272, 683–698 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Forrester P.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  35. 35.
    Forrester P.: Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry. Nonlinearity 19, 2989–3002 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Forrester P., Mays A.: Finite size corrections in random matrix theory and Odlyzko’s data set for the Riemann zeros. Proc. R. Soc. A 471, 20150436 (2015)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Hastings S., McLeod J.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation. Arch. Ration. Mech. Anal. 73, 31–51 (1980)CrossRefzbMATHGoogle Scholar
  38. 38.
    Its A., Izergin A., Korepin V., Slavnov N.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Its A., Krasovsky I.: Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump. Contemp. Math. 458, 215–247 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, (2008)Google Scholar
  41. 41.
    Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Johansson K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153, 259–296 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Krasovsky I.: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. 25, 1249–1272 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    NIST Digital Library of Mathematical Functions.
  45. 45.
    Okounkov, A.: Random matrices and random permutations. Int. Math. Res. Not. 2000(20), 1043–1095 (2000)Google Scholar
  46. 46.
    Prähofer M., Spohn H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)ADSCrossRefGoogle Scholar
  47. 47.
    Takeuchi K., Sano M.: Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    The Coulomb fluid and the fifth Painleve transendent. Chen Ning Yang: A great physicist of the twentieth century. In: Liu, C., Yau, S.-T. (eds.). pp. 131–146, International Press, Cambridge (1995)Google Scholar
  49. 49.
    Tracy C., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tracy C., Widom H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Widom H.: Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals. Commun. Math. Phys. 171, 159–180 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations