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

SIGMA 14 (2018), 018, 43 pages      arXiv:1708.02519      https://doi.org/10.3842/SIGMA.2018.018
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

Christophe Charlier a and Alfredo Deaño b
a) Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, SE-114 28 Stockholm, Sweden
b) School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK

Received November 02, 2017, in final form February 27, 2018; Published online March 07, 2018

Abstract
We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation.

Key words: asymptotic analysis; Riemann-Hilbert problems; Hankel determinants; random matrix theory; Painlevé equations.

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References

  1. Anderson G.W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Vol. 118, Cambridge University Press, Cambridge, 2010.
  2. Atkin M., Charlier C., Zohren S., On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble, Nonlinearity 31 (2018), 1155-1196, arXiv:1611.00631.
  3. Bertola M., Bothner T., Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity, Int. Math. Res. Not. 2015 (2015), 9330-9399, arXiv:1401.1408.
  4. Bertola M., Lee S.Y., First colonization of a hard-edge in random matrix theory, Constr. Approx. 31 (2010), 231-257, arXiv:0711.3625.
  5. Bohigas O., Pato M.P., Missing levels in correlated spectra, Phys. Lett. B 595 (2004), 171-176, nucl-th/0403006.
  6. 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, Comm. Math. Phys. 337 (2015), 1397-1463, arXiv:1407.2910.
  7. Brézin E., Hikami S., Characteristic polynomials of real symmetric random matrices, Comm. Math. Phys. 223 (2001), 363-382, math-ph/0103012.
  8. Buckingham R., Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials, arXiv:1706.09005.
  9. Charlier C., Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities, Int. Math. Res. Not., to appear, arXiv:1706.03579.
  10. Charlier C., Claeys T., Asymptotics for Toeplitz determinants: perturbation of symbols with a gap, J. Math. Phys. 56 (2015), 022705, 23 pages, arXiv:1409.0435.
  11. Charlier C., Claeys T., Thinning and conditioning of the circular unitary ensemble, Random Matrices Theory Appl. 6 (2017), 1750007, 51 pages, arXiv:1604.08399.
  12. Claeys T., Birth of a cut in unitary random matrix ensembles, Int. Math. Res. Not. 2008 (2008), rnm166, 40 pages, arXiv:0711.2609.
  13. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
  14. Clarkson P.A., Jordaan K., The relationship between semiclassical Laguerre polynomials and the fourth Painlevé equation, Constr. Approx. 39 (2014), 223-254, arXiv:1301.4134.
  15. Deaño A., Simm N.J., On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials, J. Approx. Theory 220 (2017), 44-59, arXiv:1610.08561.
  16. 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.
  17. Deift P., Its A., Krasovsky I., Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math. 174 (2011), 1243-1299, arXiv:0905.0443.
  18. 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.
  19. 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.
  20. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 119-123, math.AP/9201261.
  21. 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.
  22. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
  23. Forrester P.J., Witte N.S., Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys. 219 (2001), 357-398, math-ph/0103025.
  24. Foulquié Moreno A., Martínez-Finkelshtein A., Sousa V.L., On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials, J. Approx. Theory 162 (2010), 807-831, arXiv:0905.2753.
  25. Garoni T.M., On the asymptotics of some large Hankel determinants generated by Fisher-Hartwig symbols defined on the real line, J. Math. Phys. 46 (2005), 043516, 19 pages, math-ph/0411019.
  26. Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
  27. Its A., Krasovsky I., Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump, in Integrable systems and random matrices, Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 215-247, arXiv:0706.3192.
  28. Kajiwara K., Ohta Y., Determinant structure of the rational solutions for the Painlevé IV equation, J. Phys. A: Math. Gen. 31 (1998), 2431-2446, solv-int/9709011.
  29. Krasovsky I., Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant, Duke Math. J. 139 (2007), 581-619, math-ph/0411016.
  30. Kuijlaars A.B.J., McLaughlin K.T.-R., Van Assche W., Vanlessen M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$, Adv. Math. 188 (2004), 337-398, math.CA/0111252.
  31. Kuijlaars A.B.J., Vanlessen M., Universality for eigenvalue correlations at the origin of the spectrum, Comm. Math. Phys. 243 (2003), 163-191, math-ph/0305044.
  32. Mehta M.L., Random matrices, Pure and Applied Mathematics (Amsterdam), Vol. 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004.
  33. Mehta M.L., Normand J.-M., Probability density of the determinant of a random Hermitian matrix, J. Phys. A: Math. Gen. 31 (1998), 5377-5391.
  34. Mo M.Y., The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the ''birth of a cut'' transition, Int. Math. Res. Not. 2008 (2008), rnn042, 51 pages, arXiv:0711.3208.
  35. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{{\rm II}}$ and $P_{{\rm IV}}$, Math. Ann. 275 (1986), 221-255.
  36. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V. (Editors), NIST digital library of mathematical functions, Release 1.0.13 of 2016-09-16, available at http://dlmf.nist.gov/.
  37. Saff E.B., Totik V., Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Berlin, 1997.
  38. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  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.
  40. Winternitz P., Physical applications of Painlevé type equations quadratic in the highest derivatives, in Painlevé Transcendents (Sainte-Adèle, PQ, 1990), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, Plenum, New York, 1992, 425-431.
  41. Wu X.-B., Xu S.-X., Zhao Y.-Q., Gaussian unitary ensemble with boundary spectrum singularity and $\sigma$-form of the Painlevé II equation, Stud. Appl. Math. 140 (2018), 221-251, arXiv:1706.03174.

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