Asymptotics of a cubic sine kernel determinant. (English) Zbl 1318.82015

St. Petersbg. Math. J. 26, No. 4, 515-565 (2015) and Algebra Anal. 26, No. 4, 2-91 (2014).
Summary: 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 \( \gamma \).


82B23 Exactly solvable models; Bethe ansatz
33E05 Elliptic functions and integrals
34E05 Asymptotic expansions of solutions to ordinary differential equations
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
82C23 Exactly solvable dynamic models in time-dependent statistical mechanics
15B52 Random matrices (algebraic aspects)
15A15 Determinants, permanents, traces, other special matrix functions
Full Text: DOI arXiv


[1] [BT] E. Basor and C. Tracy, Some problems associated with the asymptotics of \(\tau \)-functions, Surikagaku 30 (1992), no. 3, 71-76.
[2] [BW] E. Basor and H. Widom, Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols, J. Funct. Anal. 50 (1983), no. 3, 387-413. · Zbl 0509.47020
[3] [BE] H. Bateman and A. Erdelyi, Higher transcendental functions, McGraw-Hill, New York, 1953. · Zbl 0143.29202
[4] [Ber] M. Bertola, On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation, Nonlinearity 25 (2012), no. 4, 1179-1185. · Zbl 1254.34123
[5] [BWieg] B. Bettelheim and P. Wiegmann, Fermi distribution of semiclassical non-equlibirum Fermi states, Phys. Rev. B, 085102 (2011). · Zbl 1222.82056
[6] [BB] 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; English transl.; St. Petersburg Math. J. 7 (1996), no. 6, 925-942. · Zbl 0862.35148
[7] [BI1] P. Bleher and A. 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. · Zbl 0956.42014
[8] [BI2] \bysame, Double scaling limit in the random matrix model: the Riemann-Hilbert approach, Comm. Pure Appl. Math. 56 (2003), no. 4, 433-516. · Zbl 1032.82014
[9] [BoI1] T. Bothner and A. Its, Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models, Commun. Math. Phys. 328 (2014), no. 1, 155-202. · Zbl 1290.15004
[10] [BoI2] \bysame, The nonlinear steepest descent approach to the singular asymptotics of the second Painlev\'e transcendent, Phys. D 241 (2012), no. 23-24, 2204-2225. · Zbl 1266.34142
[11] [CK] T. Claeys and A. Kuijlaars, Universality of the double scaling limit in random matrix models, Comm. Pure Appl. Math. 59 (2006), no. 11, 1573-1603. · Zbl 1111.35031
[12] [D] P. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Math., vol. 3, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0997.47033
[13] [DIK] P. Deift, A. Its, and I. Krasovsky, On asymptotics of a Toeplitz determinant with singularities, arXiv:1206.1292 (2012). · Zbl 1326.35218
[14] [DIKZ] 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. · Zbl 1116.15019
[15] [DIZ] P. Deift, A. Its, and X. 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. · Zbl 0936.47028
[16] [DKM] 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. · Zbl 0918.31001
[17] [DKMVZ] 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. · Zbl 0944.42013
[18] [DZ1] P. A. 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. · Zbl 0771.35042
[19] [dyson] F. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47 (1976), no. 2, 171-183. · Zbl 0323.33008
[20] [E] T. Ehrhardt, Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Comm. Math. Phys. 262 (2006), no. 2, 317-341. · Zbl 1113.82030
[21] [FT] L. D. Faddeev and L. A. Takhtadzhyan, The Hamiltonian approach in solution theory, Nauka, Moscow, 1986; English transl., Springer-Verlag, Berlin, 1987.
[22] [FIKN] A. Fokas, A. Its, A. Kapaev, and V. Novokshenov, Painlev\'e transcendents. The Riemann-Hilbert approach, Math. Surveys and Monogr., vol. 128, American Math. Soc., Providence, RI, 2006. · Zbl 1111.34001
[23] [FN] H. Flaschka and A. C. Newell, Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), no. 1, 65-116. · Zbl 0439.34005
[24] [IIK] A. Its, A. Izergin, and V. Korepin, Long-distance asymptotics of temperature correlators of the impenetrable Bose gas, Comm. Math. Phys. 130 (1990), no. 3, 471-488. · Zbl 0702.76089
[25] [IIKS] A. Its, A. Izergin, V. Korepin, and N. Slavnov, Differential equations for quantum correlation functions, Intern. J. Modern. Phys. B 4 (1990), no. 5, 1003-1037. · Zbl 0719.35091
[26] [IIKV1] A. Its, A. Izergin, and V. Korepin, Large time and distance asymptotics of the temperature field correlator in the impenetrable Bose gas, Nucl. Phys. B 348 (1991), no. 3, 757-765. · Zbl 0743.35076
[27] [IIKV2] A. Its, A. Izergin, V. Korepin,and 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. · Zbl 0800.35034
[28] [IK] A. Its and I. Krasovsky, Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump, Contemp. Math. 458 (2008), 215-247. · Zbl 1163.15027
[29] [JMU] M. Jimbo, T. Miwa, and K. 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. · Zbl 1194.34167
[30] [KKMST] N. Kitanine, K. Kozlowski, J. Maillet, N. Slavnov, and V. Terras, Riemann-Hilbert approach to a generalised sine kernel and applications, Comm. Math. Phys. 291 (2009), no. 3, 691-761. · Zbl 1189.45018
[31] [K] I. 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. 2004, no. 25, 1249-1272. · Zbl 1077.60079
[32] [M] M. L. Mehta, Random Matrices, 2nd ed., Acad. Press, Boston, MA, 1991.
[33] [MT1] B. M. McCoy and Sh. Tang, Connection formulae for Painlev\'e V functions, Phys. D 19 (1986), no. 1, 42-72. · Zbl 0639.58041
[34] [MT2] \bysame, Connection formulae for Painlev\'e functions. II, The \(\delta \)-function Bose gas problem, Phys. D 20 (1986), no. 2-3, 187-216. · Zbl 0656.35114
[35] [Mo] H. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number Theory (Proc. Sympos. Pure Math., St. Louis Univ., 1972), vol. 24, Amer. Math. Soc, Providence, RI, 1973, pp. 181-193.
[36] [O] A. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273-308. · Zbl 0615.10049
[37] [PS] 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. · Zbl 0916.15009
[38] [SA] H. Segur and M. Ablowitz, Asymptotic solutions of the Korteweg de Vries equation, Stud. Appl. Math. 57 (1977), no. 1, 13-44. · Zbl 0369.35055
[39] [S] B. Simon, Trace ideals and their applications, London Math. Soc. Lecture Note Ser. vol. 35, Cambridge Univ. Press, Cambridge-New York, 1979. · Zbl 0423.47001
[40] [suleiman] B. I. Suleimanov, On asymptotics of regular solutions for a special kind of Painlev\'e V equation, Lecture Notes in Math., vol. 1193, Springer-Verlag, Berlin, 1986, pp. 230-260.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.