Asymptotics of a cubic sine kernel determinant
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- by T. Bothner and A. Its
- St. Petersburg Math. J. 26 (2015), 515-565
- DOI: https://doi.org/10.1090/spmj/1350
- Published electronically: May 6, 2015
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Abstract:
The one-parameter family of Fredholm determinants $\det (I-\gamma K_{\mathrm {csin}})$, $\gamma \in \mathbb {R}$, is studied for an integrable Fredholm operator $K_{\mathrm {csin}}$ that acts on the interval $(-s,s)$ and whose kernel is a cubic generalization of the sine kernel that appears in random matrix theory. This Fredholm determinant arises in the description of the Fermi distribution of semiclassical nonequilibrium Fermi states in condensed matter physics as well as in the random matrix theory. By using the Riemann–Hilbert method, the large $s$ asymptotics of $\det (I-\gamma K_{\mathrm {csin}} )$ is calculated for all values of the real parameter $\gamma$.References
- E. Basor and C. Tracy, Some problems associated with the asymptotics of $\tau$-functions, Surikagaku 30 (1992), no. 3, 71–76.
- Estelle Basor and Harold Widom, Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols, J. Funct. Anal. 50 (1983), no. 3, 387–413. MR 695420, DOI 10.1016/0022-1236(83)90010-1
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- M. Bertola, On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation, Nonlinearity 25 (2012), no. 4, 1179–1185. MR 2904274, DOI 10.1088/0951-7715/25/4/1179
- B. Bettelheim and P. Wiegmann, Fermi distribution of semiclassical non-equlibirum Fermi states, Phys. Rev. B, 085102 (2011).
- A. M. Budylin and V. S. Buslaev, Quasiclassical asymptotics of the resolvent of an integral convolution operator with a sine kernel on a finite interval, Algebra i Analiz 7 (1995), no. 6, 79–103 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 6, 925–942. MR 1381979
- Pavel Bleher and Alexander Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324, DOI 10.2307/121101
- Pavel Bleher and Alexander Its, Double scaling limit in the random matrix model: the Riemann-Hilbert approach, Comm. Pure Appl. Math. 56 (2003), no. 4, 433–516. MR 1949138, DOI 10.1002/cpa.10065
- Thomas Bothner and Alexander Its, Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models, Comm. Math. Phys. 328 (2014), no. 1, 155–202. MR 3196983, DOI 10.1007/s00220-014-1950-z
- Thomas Bothner and Alexander Its, The nonlinear steepest descent approach to the singular asymptotics of the second Painlevé transcendent, Phys. D 241 (2012), no. 23-24, 2204–2225. MR 2998123, DOI 10.1016/j.physd.2012.02.014
- Tom Claeys and Arno B. J. Kuijlaars, Universality of the double scaling limit in random matrix models, Comm. Pure Appl. Math. 59 (2006), no. 11, 1573–1603. MR 2254445, DOI 10.1002/cpa.20113
- P. A. Deift, 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; American Mathematical Society, Providence, RI, 1999. MR 1677884
- P. Deift, A. Its, and I. Krasovsky, Eigenvalues of Toeplitz matrices in the bulk of the spectrum, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 4, 437–461. MR 3077466
- P. Deift, A. Its, I. Krasovsky, and X. Zhou, The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math. 202 (2007), no. 1, 26–47. MR 2301810, DOI 10.1016/j.cam.2005.12.040
- Percy A. Deift, Alexander R. Its, and Xin Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2) 146 (1997), no. 1, 149–235. MR 1469319, DOI 10.2307/2951834
- P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), no. 3, 388–475. MR 1657691, DOI 10.1006/jath.1997.3229
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, 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), no. 11, 1335–1425. MR 1702716, DOI 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
- P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
- Freeman J. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), no. 2, 171–183. MR 406201
- Torsten Ehrhardt, Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Comm. Math. Phys. 262 (2006), no. 2, 317–341. MR 2200263, DOI 10.1007/s00220-005-1493-4
- L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 905674, DOI 10.1007/978-3-540-69969-9
- Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlevé transcendents, Mathematical Surveys and Monographs, vol. 128, American Mathematical Society, Providence, RI, 2006. The Riemann-Hilbert approach. MR 2264522, DOI 10.1090/surv/128
- Hermann Flaschka and Alan C. Newell, Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), no. 1, 65–116. MR 588248
- A. R. It⋅s, A. G. Izergin, and V. E. Korepin, Long-distance asymptotics of temperature correlators of the impenetrable Bose gas, Comm. Math. Phys. 130 (1990), no. 3, 471–488. MR 1060387
- A. R. It⋅s, A. G. Izergin, V. E. Korepin, and N. A. Slavnov, Differential equations for quantum correlation functions, Proceedings of the Conference on Yang-Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, 1990, pp. 1003–1037. MR 1064758, DOI 10.1142/S0217979290000504
- A. R. It⋅s, A. G. Izergin, and V. E. Korepin, Large time and distance asymptotics of the temperature field correlator in the impenetrable Bose gas, Nuclear Phys. B 348 (1991), no. 3, 757–765. MR 1083921, DOI 10.1016/0550-3213(91)90213-H
- A. R. It⋅s, A. G. Izergin, V. E. Korepin, and G. G. Varzugin, Large time and distance asymptotics of field correlation function of impenetrable bosons at finite temperature, Phys. D 54 (1992), no. 4, 351–395. MR 1147598, DOI 10.1016/0167-2789(92)90043-M
- A. Its and I. Krasovsky, Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump, Integrable systems and random matrices, Contemp. Math., vol. 458, Amer. Math. Soc., Providence, RI, 2008, pp. 215–247. MR 2411909, DOI 10.1090/conm/458/08938
- Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), no. 2, 306–352. MR 630674, DOI 10.1016/0167-2789(81)90013-0
- N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras, Riemann-Hilbert approach to a generalised sine kernel and applications, Comm. Math. Phys. 291 (2009), no. 3, 691–761. MR 2534790, DOI 10.1007/s00220-009-0878-1
- I. V. Krasovsky, 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 (2004), 1249–1272. MR 2047176, DOI 10.1155/S1073792804140221
- Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
- Barry M. McCoy and Shuang Tang, Connection formulae for Painlevé $\textrm {V}$ functions, Phys. D 19 (1986), no. 1, 42–72. MR 840027, DOI 10.1016/0167-2789(86)90053-9
- Barry M. McCoy and Shuang Tang, Connection formulae for Painlevé functions. II. The $\delta$ function Bose gas problem, Phys. D 20 (1986), no. 2-3, 187–216. MR 859353, DOI 10.1016/0167-2789(86)90030-8
- H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
- A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. MR 866115, DOI 10.1090/S0025-5718-1987-0866115-0
- L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), no. 1-2, 109–147. MR 1435193, DOI 10.1007/BF02180200
- M. J. Ablowitz and H. Segur, Asymptotic solutions of the Korteweg-deVries equation, Studies in Appl. Math. 57 (1976/77), no. 1, 13–44. MR 481656, DOI 10.1002/sapm197757113
- Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149
- B. I. Suleimanov, On asymptotics of regular solutions for a special kind of Painlevé V equation, Lecture Notes in Math., vol. 1193, Springer-Verlag, Berlin, 1986, pp. 230–260.
Bibliographic Information
- T. Bothner
- Affiliation: Centre de recherchrs mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour, Montréal, Québec H3T 1J4, Canada
- ORCID: 0000-0001-8300-7467
- Email: bothner@crm.umontreal.ca
- A. Its
- Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, Indiana 46202
- Email: itsa@math.iupui.edu
- Received by editor(s): July 10, 2013
- Published electronically: May 6, 2015
- Additional Notes: This work was supported in part by the National Science Foundation (NSF) Grant DMS-1001777 and by the SPbGU grant N11.38.215.2014
- © Copyright 2015 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 515-565
- MSC (2010): Primary 82B23; Secondary 33E05, 34E05, 34M50
- DOI: https://doi.org/10.1090/spmj/1350
- MathSciNet review: 3289185
Dedicated: To the memory of Vladimir Savelievich Buslaev