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Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications

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Abstract

We investigate the asymptotic behaviour of a generalised sine kernel acting on a finite size interval [−q ; q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener–Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.

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References

  1. Akhiezer N.I. (1964) The continuous analogues of some theorems on Toeplitz matrices. Ukrainian Math. J. 16: 445–462

    Article  MATH  Google Scholar 

  2. Barnes E.W. (1901) The theory of the double gamma function. Philos. Trans. Roy. Soc. London, Ser. A 196: 265–388

    Article  ADS  Google Scholar 

  3. Barnes E.W. (1900) Genesis of the double gamma function. Proc. London Math. Soc. 31: 358–381

    Article  Google Scholar 

  4. Basor E.L., Tracy C.A. (1992) Some problems associated with the asymptotics of τ functions. Suaikaguku 30(3): 71–76

    Google Scholar 

  5. Bogoliubov N.M., Izergin A.G., Korepin V.E. (1986) Critical exponents for integrable models. Nucl. Phys. B. 275: 687

    Article  ADS  MathSciNet  Google Scholar 

  6. Bogoliubov, N.M., Izergin, A.G., Korepin V.E.: Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz. Cambridge monograph on mathematical physics, Cambridge: Cambridge Univ. press, 1993

  7. Budylin A.M., Buslaev V.S. (1995) Quasiclassical asymptotics of the resolvent of an integral convolution operator with a sine kernel on a finite interval. Alg. i Analiz. 7(6): 79–103

    MATH  MathSciNet  Google Scholar 

  8. Cheianov V.V., Zvonarev M.R. (2004) Zero temperature correlation functions for the impenetrable fermion gas. J. Phys. A:Math. Gen. 37: 2261–2297

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Colomo F., Izergin A.G., Korepin V.E., Tognetti V. (1992) Correlators in the Heisenberg XX0 chain as Fredholm determinants. Phys. Lett. A 169: 237–247

    Article  ADS  MathSciNet  Google Scholar 

  10. Colomo F., Izergin A.G., Korepin V.E., Tognetti V. (1993) Temperature correlation functions in the XX0 Heisenberg chain. Teor. Mat. Fiz. 94: 19–38

    MathSciNet  Google Scholar 

  11. Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. In: Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Berlin: Springer, 1993, pp. 181–204

  12. Deift, P.A., Zhou, X.: Long-Time Behaviour of the Non-focusing Nonlinear Schrödinger Equation - a Case Study. Lectures in Mathematical Sciences, Vol. 5, Tokyo: University of Tokyo, 1994

  13. Deift P.A., Its A.R., Zhou X. (1997) A Riemann–Hilbert approach to asymptotics problems arising in the theory of random matrix models and also in the theory of integrable statistical mechanics. Ann. Math. 146: 149–235

    Article  MATH  MathSciNet  Google Scholar 

  14. Deift P.A., Zhou X. (1997) A steepest descent method for oscillatory Riemann–Hilbert problems. Intl. Math. Res. 6: 285–299

    Google Scholar 

  15. Deift P.A., Its A.R., Krasovsky I., Zhou X. (2007) The Widom-Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202(1): 26–47

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Deift, P., Its, A.R., Krasovsky, I.: Toeplitz and Hankel determinants with singularities: announcement of results, http://arXiv.org/abs/0809.2420v1[math.FA], 2008

  17. Deift, P., Its, A.R., Krasovsky, I.: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities, http://arXiv.org/abs/0905.0443v1[math.FA], 2009

  18. des Cloizeaux J., Mehta M.L. (1973) Asymptotic behaviour of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14: 1648–1650

    Article  MATH  ADS  Google Scholar 

  19. Dyson F. (1976) Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47: 171–183

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Ehrhardt T. (2006) Dyson’s constant in the asymptotics of the fredholm determinant of the sine kernel. Commun. Math. Phys. 262: 317–341

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Gaudin M. (1961) Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire. Nucl. Phys. 25: 447–458

    Article  MATH  Google Scholar 

  22. Gaudin M., Mehta M.L. (1960) On the density of eigenvalues of a random matrix. Nucl. Phys 18: 420–427

    Article  MathSciNet  Google Scholar 

  23. Haldane F.D.M. (1980) General relation of correlation exponents and spectral properties of one-dimensional Fermi systems: Application to the anisotropic s = 1/2 Heisenberg chain. Phys. Rev. Lett. 45: 1358

    Article  ADS  MathSciNet  Google Scholar 

  24. Haldane F.D.M. (1981) Demonstration of the “Luttinger liquid” character of Bethe-ansatz soluble models of 1-D quantum fluids. Phys. Lett. A 81: 153

    Article  ADS  MathSciNet  Google Scholar 

  25. Haldane F.D.M. (1981) Luttinger liquid theory of one-dimensional quantum fluids: I. Properties of the Luttinger model and their extension to the general interacting 1D spinless Fermi gas. J. Phys. C: Solid State Phys. 14: 2585

    Article  ADS  Google Scholar 

  26. Its A.R. (1981) Asymptotic behaviour of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations. Dokl. Akad. Nauk SSSR 261(1): 14–18

    MathSciNet  Google Scholar 

  27. Its A.R., Izergin A.G., Korepin V.E. (1990) Long-distance asymptotics of temperature correlators of the impenetrable Bose gas. Commun. Math. Phys. 130: 471–488

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Its A.R., Izergin A.G., Korepin V.E. (1990) Temperature correlators of the impenetrable Bose gas as an integrable system. Commun. Math. Phys. 129: 205–222

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A. (1993) Temperature correlations of quantum spins. Phys. Rev. Lett. 70: 1704

    Article  ADS  Google Scholar 

  30. Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A. (1990) Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4: 1003–1037

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Its A.R., Izergin A.G., Korepin V.E., Varguzin G.G. (1991) Large time and distance asymptotics of the correlator of the impenetrable bosons at finite temperature. Physica D 54: 351

    Article  ADS  Google Scholar 

  32. Its, A.R., Krasovsky, I.: Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump. Contemp. Maths. (AMS series) 458, Providence, RI: Amer. Math. Soc., 2008 pp. 215–247

  33. Jimbo M., Miwa T., Mori Y., Sato M. (1980) Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D 1: 80–158

    Article  ADS  MathSciNet  Google Scholar 

  34. Kac M. (1954) Toeplitz matrices, translation kernels and related problem in probability. Duke Math. J. 21: 501–510

    Article  MATH  MathSciNet  Google Scholar 

  35. Kitanine, N., Kozlowski, K.K., Maillet, J.-M., Slavnov, N.A., Terras, V.: Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions. J. Stat. Mech. (2009) P04003, doi:10.1088/1742-5468/2009/04/R04003

  36. Korepin V.E., Slavnov N.A. (1990) The time dependent correlation function of an impenetrable Bose gas as a Fredholm minor. I. Commun. Math. Phys. 129(1): 103–113

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Kozlowski, K.K.: Truncated Wiener-Hopf operators with Fischer-Hartwig singularities. http://arXiv.org/abs/0805.3902v1[math.FA], 2008

  38. Krasovsky I.V. (2004) 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

    Article  MATH  MathSciNet  Google Scholar 

  39. Kuijlaars A.B.J., McLaughlin K.T.-R., Van Assche W., Vanlessen M. (2004) The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]. Adv. in Math. 188: 337–398

    Article  MATH  MathSciNet  Google Scholar 

  40. Lenard A. (1964) Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5: 930–943

    Article  ADS  MathSciNet  Google Scholar 

  41. Lenard A. (1966) One-dimensional impenetrable bosons in thermal equilibrium. J. Math. Phys. 7: 1268–1272

    Article  ADS  MathSciNet  Google Scholar 

  42. McCoy B.M., Tang S. (1986) Connection formulae for Painlevé V functions. II. The delta function Bose gas problem. Physica D 20: 187–216

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. McCoy B.M., Perk J.H.H., Shrock R.E. (1983) Time-dependent correlation functions of the transverse Ising chain at the critical magnetic field. Nucl. Phys. B 220: 35–47

    Article  ADS  MathSciNet  Google Scholar 

  44. Widom H. (1971) The strong Szegö limit theorem for circular arcs. Indiana Univ. Math. J. 21: 277–283

    Article  MATH  MathSciNet  Google Scholar 

  45. Widom H. (1994) The asymptotics of a continuous analogue of orthogonal polynomials. J. Approx. Th. 77: 51–64

    Article  MATH  MathSciNet  Google Scholar 

  46. Widom H. (1995) Asymptotics for the Fredholm determinant of the Sine Kernel on a Union of Intervals. Commun. Math. Phys. 171: 159–180

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. LPTM, Université de Cergy-Pontoise et CNRS, Cergy-Pontoise, France

    N. Kitanine

  2. Laboratoire de Physique, ENS Lyon et CNRS, Lyon, France

    K. K. Kozlowski, J. M. Maillet & V. Terras

  3. Steklov Mathematical Institute, Moscow, Russia

    N. A. Slavnov

Authors
  1. N. Kitanine
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  2. K. K. Kozlowski
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  3. J. M. Maillet
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  4. N. A. Slavnov
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  5. V. Terras
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Corresponding author

Correspondence to J. M. Maillet.

Additional information

Communicated by L. Takhtajan

On leave of absence from: LPTA, Université Montpellier II et CNRS, France

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Kitanine, N., Kozlowski, K.K., Maillet, J.M. et al. Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications. Commun. Math. Phys. 291, 691–761 (2009). https://doi.org/10.1007/s00220-009-0878-1

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  • DOI: https://doi.org/10.1007/s00220-009-0878-1

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