Charlier, Christophe
[UCL]
We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher–Hartwig singularities. This generalises two results: (1) a result of Berestycki, Webb, and Wong [5] for root-type singularities and (2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomial of the associated conditional point process, can be expressed in terms of these determinants.
- Akemann, The Oxford Hand of Random Matrix Theory (2011)
-
Anderson,
G. W.
, A.Guionnet, and O.Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge:
Cambridge University Press, 2010.
- Basor Estelle, Asymptotic formulas for Toeplitz determinants, 10.1090/s0002-9947-1978-0493480-x
- Basor E., , 10.1512/iumj.1979.28.28070
- Berestycki, Random Hermitian Matrices and Gaussian Multiplicative Chaos
- Berggren Tomas, Duits Maurice, Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble, 10.1007/s11040-017-9250-4
- M. Bleher Pavel, Its Alexander, Asymptotics of the partition function of a random matrix model, 10.5802/aif.2147
- Bohigas O., Pato M.P., Missing levels in correlated spectra, 10.1016/j.physletb.2004.05.065
- Bohigas, Phys. Rev. E, 74, no. (2006)
- Bothner, Non-oscillatory Asymptotics, 04462 (1702)
- Bothner Thomas, Deift Percy, Its Alexander, Krasovsky Igor, On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I, 10.1007/s00220-015-2357-1
- Bothner, On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II, 213 (2017)
- Böttcher Albrecht, Silbermann Bernd, Toeplitz Operators and Determinants Generated by Symbols with One Fisher-Hartwig Singularity, 10.1002/mana.19861270108
- Charlier Christophe, Claeys Tom, Asymptotics for Toeplitz determinants: Perturbation of symbols with a gap, 10.1063/1.4908105
- Charlier Christophe, Claeys Tom, Thinning and conditioning of the circular unitary ensemble, 10.1142/s2010326317500071
- Charlier, Asymptotics for Hankel determinants associated to a Hermite weight with a varying discontinuity
- Claeys, SIGMA Symmetry Integrability Geom. Methods Appl, 12, 44 (2016)
- Claeys T., Grava T., McLaughlin K. D. T.-R., Asymptotics for the Partition Function in Two-Cut Random Matrix Models, 10.1007/s00220-015-2412-y
- Claeys T., Krasovsky I., Toeplitz determinants with merging singularities, 10.1215/00127094-3164897
- Daley, An Introduction to the Theory of Point Processes: Volume II: General Theory and structure. (2008)
- Deaño Alfredo, Simm Nicholas J., On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials, 10.1016/j.jat.2017.04.004
- Deift, Amer. Math. Soc, 3 (2000)
- Deift Percy, Its Alexander, Krasovsky Igor, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, 10.4007/annals.2011.174.2.12
- Deift, On the Asymptotics of a Toeplitz Determinant With singularities (2014)
- Deift P, Kriecherbauer T, McLaughlin K.T.-R, New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field, 10.1006/jath.1997.3229
- 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, 10.1002/(sici)1097-0312(199911)52:11<1335::aid-cpa1>3.0.co;2-1
- Deift, Comm. Pure Appl. Math, 52, 1491 (1999)
- Deift P., Zhou X., A steepest descent method\\ for oscillatory Riemann-Hilbert problems, 10.1090/s0273-0979-1992-00253-7
- Deift P., Zhou X., A Steepest Descent Method for Oscillatory Riemann--Hilbert Problems. Asymptotics for the MKdV Equation, 10.2307/2946540
- Ehrhardt, Operator Theory: Adv. Appl., 124, 217 (2001)
- Ercolani N., McLaughlin K., 10.1155/s1073792803211089
- Fisher, Advan. Chem. Phys, 15, 333 (1968)
- Fokas A. S., Its A. R., Kitaev A. V., The isomonodromy approach to matric models in 2D quantum gravity, 10.1007/bf02096594
- Forrester P. J., Frankel N. E., Applications and generalizations of Fisher–Hartwig asymptotics, 10.1063/1.1699484
- 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, 10.1016/j.jat.2009.08.006
- Gakhov,
Boundary Value Problems. Oxford: Pergamon Press, 1966. Reprinted by (1990)
- Its A., Krasovsky I., Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump, 10.1090/conm/458/08938
- Johansson Kurt, On fluctuations of eigenvalues of random Hermitian matrices, 10.1215/s0012-7094-98-09108-6
- Johansson, 1 (2006)
- Kallenberg, Inst. Stat. Mimeo Ser. (1974)
- Keating J. P., Snaith N. C., Random Matrix Theory and ζ(1/2+ it), 10.1007/s002200000261
- Krasovsky I. V., Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant, 10.1215/s0012-7094-07-13936-x
- 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], 10.1016/j.aim.2003.08.015
- Lambert
- Mehta,
Random matrices,
3rd ed. Pure and Applied Mathematics Series
(2004)
- Olver, NIST Handbook of Mathematical Functions. (2010)
- Saff Edward B., Totik Vilmos, Logarithmic Potentials with External Fields, ISBN:9783642081736, 10.1007/978-3-662-03329-6
- Soshnikov A, Determinantal random point fields, 10.1070/rm2000v055n05abeh000321
- Szegő,
Orthogonal Polynomials
(1959)
- Widom Harold, The Strong Szego Limit Theorem for Circular Arcs, 10.1512/iumj.1972.21.21022
- Widom Harold, Toeplitz Determinants with Singular Generating Functions, 10.2307/2373789
- Wu Xiao-Bo, Xu Shuai-Xia, Zhao Yu-Qiu, Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ-Form of the Painlevé II Equation : Gaussian Unitary Ensemble with Boundary Spectrum Singularity, 10.1111/sapm.12197
Bibliographic reference |
Charlier, Christophe. Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities. In: International Mathematics Research Notices, Vol. rny009, no./, p. / (2018) |
Permanent URL |
http://hdl.handle.net/2078.1/196034 |