On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential. I. (English) Zbl 1321.82027

The authors study the determinant of the integral Fredholm’s operator with kernel \(\sin x(a-b) / \pi(a-b)\) which appears in the analysis of a log-gas of interacting particles. The main result is obtained in the form of a general expression for the logarithm of this determinant. The proof is lengthy and works via Riemann-Hilbert analysis.


82C22 Interacting particle systems in time-dependent statistical mechanics
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)


Full Text: DOI arXiv


[1] Basor, E.; Widom, H., Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols, J. Funct. Anal., 50, 387-413, (1983) · Zbl 0509.47020
[2] Bateman H., Erdelyi A.: Higher Transcendental Functions. McGraw-Hill, NY (1953) · Zbl 0143.29202
[3] Bleher, P.; Liechty, K., Exact solution of the six-vertex model with domain wall-boundary conditions. anti-ferroelectric phase, Commun. Pure Appl. Math., 63, 779-829, (2010) · Zbl 1192.82015
[4] Bohigas, O., Pato, M.: Randomly incomplete spectra and intermediate statistics. Phys. Rev. E 74, 036212 (2006)
[5] 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 II, In preparation · Zbl 1382.82014
[6] Budylin, A.; Buslaev, V., Quasiclassical asymptotics of the resolvent of an integral convolution operator with a sine kernel on a finite interval, Algebra i Analiz, 7, 79-103, (1995) · Zbl 0862.35148
[7] Claeys, T., Krasovsky, I.: Toeplitz determinants with merging singularities, preprint: arXiv:1403.3639 · Zbl 1333.15018
[8] Deift, P.; Its, A.; Krasovsky, I., Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. Math., 174, 1243-1299, (2011) · Zbl 1232.15006
[9] Deift, P.; Its, A.; Krasovsky, I., Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. some history and some recent results, Commun. Pure Appl. Math., 66, 1360-1438, (2013) · Zbl 1292.47016
[10] Deift, P.; Its, A.; Zhou, X., 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. Math., 146, 149-235, (1997) · Zbl 0936.47028
[11] Deift, P., Its, A., Krasovsky, I., Zhou, X.: The Widom-Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202, 26-47 (2007) · Zbl 1116.15019
[12] 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, 1335-1425, (1999) · Zbl 0944.42013
[13] Deift, P.; Venakides, S.; Zhou, X., New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Notices, 6, 286-299, (1997) · Zbl 0873.65111
[14] Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. asymptotics for the mkdv equation, Ann. Math., 137, 295-368, (1993) · Zbl 0771.35042
[15] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/ · Zbl 1019.65001
[16] Dyson, F., Fredholm determinants and inverse scattering problems, Commun. Math. Phys., 47, 171-183, (1976) · Zbl 0323.33008
[17] Dyson, F.: The Coulomb fluid and the fifth Painleve transendent. In: Liu, C.S., Yau, S.-T. (eds.) Chen Ning Yang: A Great Physicist of the Twentieth Century, pp. 131-146. International Press, Cambridge (1995) (to appear) · Zbl 0936.47028
[18] Ehrhardt, T., Dysons constant in the asymptotics of the Fredholm determinant of the sine kernel, Comm. Math. Phys., 262, 317-341, (2006) · Zbl 1113.82030
[19] Farkas H.M., Kra I.: Riemann Surfaces. Springer, New York (1980) · Zbl 0475.30001
[20] Its, A.; Izergin, A.; Korepin, V.; Slavnov, N., Differential equations for quantum correlation functions, Int. J. Mod. Phys. B, 4, 1003-1037, (1990) · Zbl 0719.35091
[21] Krasovsky, I., 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, 1249-1272, (2004) · Zbl 1077.60079
[22] Khruslov, E., Kotlyarov, V.: Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations. Spectral operator theory and related topics. Adv. Soviet Math. 19, 129-180 (1994) · Zbl 0819.58015
[23] Mehta M.L.: Random Matrices, 3rd edn. Elsevier Academic Press, San Diego (2004) · Zbl 1107.15019
[24] Slepian, D., Some asymptotic expansions for prolate spheroidal functions, J. Math. Phys., 44, 99-140, (1965) · Zbl 0128.29601
[25] Soshnikov, A., Determinantal random point fields, Usp. Mat. Nauk, 55, 107-160, (2000) · Zbl 0991.60038
[26] Whittaker, E., Watson, G.: A Course of Modern Analysis, 4th edn, reprinted. Cambridge University Press, Cambridge (2000) · JFM 45.0433.02
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.