Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 093, 18 pages      arXiv:1603.04328
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail

Peter Eichelsbacher a, Thomas Kriecherbauer b and Katharina Schüler b
a) Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
b) Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Received May 31, 2016, in final form September 11, 2016; Published online September 21, 2016

We prove precise deviations results in the sense of Cramér and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which the limiting Tracy-Widom law still predicts the correct leading order behavior. Our proofs use that the determinantal point process is given by the Christoffel-Darboux kernel for an associated family of orthogonal polynomials. The necessary asymptotic information on this kernel has mostly been obtained in [Kriecherbauer T., Schubert K., Schüler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694]. In the superlarge regime these results of do not suffice and we put stronger assumptions on the point processes. The results of the present paper and the relevant parts of [Kriecherbauer T., Schubert K., Schüler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the dissertation [Schüler K., Ph.D. Thesis, Universität Bayreuth, 2015].

Key words: determinantal point process; extreme value distribution; Tracy-Widom distribution; moderate deviations; large deviations; superlarge deviations; random matrix theory; Christoffel-Darboux kernel; Riemann-Hilbert problem.

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  1. Akemann G., Baik J., Di Francesco P. (Editors), The Oxford handbook of random matrix theory, Oxford University Press, Oxford, 2011.
  2. Anderson G.W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Vol. 118, Cambridge University Press, Cambridge, 2010.
  3. Augeri F., Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails, arXiv:1502.07983.
  4. Baik J., Buckingham R., DiFranco J., Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function, Comm. Math. Phys. 280 (2008), 463-497, arXiv:0704.3636.
  5. Baik J., Deift P., McLaughlin K.T.-R., Miller P., Zhou X., Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys. 5 (2001), 1207-1250, math.PR/0112162.
  6. Ben Arous G., Dembo A., Guionnet A., Aging of spherical spin glasses, Probab. Theory Related Fields 120 (2001), 1-67.
  7. Borot G., Guionnet A., Kozlowski K.K., Large-$N$ asymptotic expansion for mean field models with Coulomb gas interaction, Int. Math. Res. Not. 2015 (2015), 10451-10524, arXiv:1312.6664.
  8. Borovkov A.A., Mogulskii A.A., On large and superlarge deviations of sums of independent random vectors under the Cramér condition. I, Theory Probab. Appl. 51 (2007), 227-255.
  9. Cramér H., Sur un nouveau théorème-limite de la théorie des probabilités, in Confér. internat. Sci. math. Univ. Genève, Théorie des probabilités. III: Les sommes et les fonctions de variables aléatoires, Actual. sci. industr., Vol. 736, Hermann, Paris, 1938, 5-23.
  10. Credner K., Eichelsbacher P., Large deviations for the largest eigenvalue of disordered bosons and disordered fermionic systems, arXiv:1503.00984.
  11. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 1999.
  12. Deift P., Gioev D., Random matrix theory: invariant ensembles and universality, Courant Lecture Notes in Mathematics, Vol. 18, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 2009.
  13. Deift P., Its A., Krasovsky I., Asymptotics of the Airy-kernel determinant, Comm. Math. Phys. 278 (2008), 643-678, math.FA/0609451.
  14. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335-1425.
  15. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
  16. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368.
  17. Dembo A., Zeitouni O., Large deviations techniques and applications, Stochastic Modelling and Applied Probability, Vol. 38, Springer-Verlag, Berlin, 2010.
  18. Döring H., Eichelsbacher P., Edge fluctuations of eigenvalues of Wigner matrices, in High Dimensional Probability VI, Progr. Probab., Vol. 66, Birkhäuser/Springer, Basel, 2013, 261-275, arXiv:1203.2115.
  19. Döring H., Eichelsbacher P., Moderate deviations for the determinant of Wigner matrices, in Limit Theorems in Probability, Statistics and Number Theory, Springer Proc. Math. Stat., Vol. 42, Springer, Heidelberg, 2013, 253-275, arXiv:1301.2915.
  20. Döring H., Eichelsbacher P., Moderate deviations for the eigenvalue counting function of Wigner matrices, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), 27-44, arXiv:1104.0221.
  21. Eichelsbacher P., Raič M., Schreiber T., Moderate deviations for stabilizing functionals in geometric probability, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), 89-128, arXiv:1010.1665.
  22. Féral D., On large deviations for the spectral measure of discrete Coulomb gas, in Séminaire de probabilités XLI, Lecture Notes in Math., Vol. 1934, Springer, Berlin, 2008, 19-49.
  23. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  24. Götze F., Venker M., Local universality of repulsive particle systems and random matrices, Ann. Probab. 42 (2014), 2207-2242, arXiv:1205.0671.
  25. Johansson K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151-204.
  26. Johansson K., Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476, math.CO/9903134.
  27. Kriecherbauer T., Schubert K., Schüler K., Venker M., Global asymptotics for the Christoffel-Darboux kernel of random matrix theory, Markov Process. Related Fields 21 (2015), 639-694, arXiv:1401.6772.
  28. Kriecherbauer T., Venker M., Edge statistics for a class of repulsive particle systems, arXiv:1501.07501.
  29. Kuijlaars A.B.J., Vanlessen M., Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 2002 (2002), 1575-1600, math-ph/0204006.
  30. Ledoux M., Rider B., Small deviations for beta ensembles, Electron. J. Probab. 15 (2010), no. 41, 1319-1343, arXiv:0912.5040.
  31. Löwe M., Merkl F., Moderate deviations for longest increasing subsequences: the upper tail, Comm. Pure Appl. Math. 54 (2001), 1488-1520.
  32. Löwe M., Merkl F., Rolles S., Moderate deviations for longest increasing subsequences: the lower tail, J. Theoret. Probab. 15 (2002), 1031-1047.
  33. Mehta M.L., Random matrices, Pure and Applied Mathematics (Amsterdam), Vol. 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004.
  34. Pastur L., Shcherbina M., Eigenvalue distribution of large random matrices, Mathematical Surveys and Monographs, Vol. 171, Amer. Math. Soc., Providence, RI, 2011.
  35. Petrov V.V., Limit theorems of probability theory. Sequences of independent random variables, Oxford Studies in Probability, Vol. 4, The Clarendon Press, Oxford University Press, New York, 1995.
  36. Schüler K., Moderate, large, and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles, Ph.D. Thesis, Universität Bayreuth, 2015, available at
  37. Soshnikov A., Determinantal random point fields, Russ. Math. Surv. 55 (2000), 923-975, math.PR/0002099.
  38. Tracy C.A., Widom H., Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174, hep-th/9210074.
  39. Vanlessen M., Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx. 25 (2007), 125-175, math.CA/0504604.

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