Journal of Statistical Physics

, Volume 170, Issue 4, pp 672–683 | Cite as

A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model

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Abstract

We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximation results for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlevé-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).

Keywords

Two-dimensional Ising model 2-Point function Short distance expansion Action integral 

Mathematics Subject Classification

Primary 82B20 Secondary 70S05 34M55 

References

  1. 1.
    Ablowitz, M., Segur, H.: Asymptotic solutions of the Korteweg de Vries equation. Stud. Appl. Math. 571, 13–44 (1977)CrossRefMATHGoogle Scholar
  2. 2.
    Bothner, T., Its, A., Prokhorov, A.: On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential. Preprint arXiv:1708.06480
  3. 3.
    Fokas, A., Its, A., Kapaev, A., Novokshenov, V.: Painlevé transcendents: the Riemann–Hilbert approach. In: Mathematical Surveys and Monographs 128. AMS, Providence (2006)Google Scholar
  4. 4.
    Gamayun, O., Iorgov, N., Lisovyy, O.: How instanton combinatorics solves Painlevé VI, V and III’s. J. Phys. A 46, 335203 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Its, A., Lisovyy, O., Tykhyy, Y.: Connection problem for the sine-Gordon/Painlevé III tau-function and irregular conformal blocks. Int. Math. Res. Notices 18, 8903–8924 (2014)MATHGoogle Scholar
  6. 6.
    Its, A., Prokhorov, A.: Connection problem for the tau-function of the sine-Gordon reduction of Painlevé-III equation via the Riemann–Hilbert approach. Int. Math. Res. Notices (2016).  https://doi.org/10.1093/imrn/rnv375
  7. 7.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica D2, 306–352 (1981)ADSMathSciNetMATHGoogle Scholar
  8. 8.
    Lebowitz, J., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys. 25, 276–282 (1972)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Lukyanov, S., Zamolodchikov, A.: Exact expectation values of local fields in quantum sine-Gordon model. Nucl. Phys. B 493, 571–587 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    McCoy, B., Tracy, C., Wu, T.: Painlevé functions of the third kind. J. Math. Phys. 18, 1058–1092 (1977)ADSCrossRefMATHGoogle Scholar
  11. 11.
    McCoy, B., Wu, T.: The Two-Dimensional Ising Model, 2nd edn. Dover Publications, Mineola (2014)MATHGoogle Scholar
  12. 12.
    NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov
  13. 13.
    Okamoto, K.: Polynomial Hamiltonians associated with Painlevé equations. I. In: Proceedings of the Japan Academy, Series A, vol. 56 (1980)Google Scholar
  14. 14.
    Palmer, J.: Planar Ising correlations. Progress in Mathematical Physics, vol. 49. Birkäuser Boston Inc., Boston (2007)MATHGoogle Scholar
  15. 15.
    Tracy, C.: Painlevé transcendents and scaling functions of the two-dimensional Ising model. In: Barut, A.O. (ed.) Nonlinear Equations in Physics and Mathematics, pp. 221–237. D. Reidel Publ. Co., Dordrecht (1978)CrossRefGoogle Scholar
  16. 16.
    Tracy, C.: Asymptotics of a \(\tau \)-function arising in the two-dimensional Ising model. Commun. Math. Phys. 142, 297–311 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tracy, C., Widom, H.: Asymptotics of a class of solutions to the cylindrical Toda equations. Commun. Math. Phys. 190, 697–721 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wu, T.-T.: Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I. Phys. Rev. 149, 380–401 (1966)ADSCrossRefGoogle Scholar
  19. 19.
    Wu, T., McCoy, B., Tracy, C., Barouch, E.: Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region. Phys. Rev. 13, 316–374 (1976)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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