Abstract
In the 1977 paper of McCoy et al. (J. Math. Phys. 18, 1058–1092, 1977) it was shown that the limiting two-point correlation function in the two-dimensional Ising model is related to a second order nonlinear Painlevé function. This result identified the scaling function as a tau-function and the corresponding connection problem was solved by Tracy (Commun. Math. Phys. 142, 297–311, 1991), see also the works by Tracy and Widom (Commun. Math. Phys. 190, 697–721, 1998). Here we present the solution to a certain generalized version of the above connection problem which is obtained through a refinement of the techniques chosen in Bothner (J. Stat. Phys. 170, 672–683, 2018).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baik, J., Buckingham, R., DiFranco, J.: Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Commun. Math. Phys. 280, 463–497 (2008)
Basor, E., Tracy, C.: Asymptotics of a tau-function and Toeplitz determinants with singular generating functions. Suppl. 1A 7, 83–107 (1992)
Bothner, T.: A short note on the scaling function constant problem in the two-dimensional Ising model. J. Stat. Phys. 170, 672–683 (2018)
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. arXiv:1708.06480
Deift, P., Its, A., Krasovsky, I.: Asymptotics of the Airy-kernel determinant. Commun. Math. Phys. 278, 643–678 (2008)
Deift, P., Its, A., Krasovsky, I., Zhou, X.: The Widom-Dyson constant and related questions of the asymptotic analysis of Toeplitz determinants, Proceedings of the AMS meeting, Atlanta 2005. J. Comput. Appl. Math. 202, 26–47 (2007)
Dyson, F.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976)
Ehrhardt, T.: Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. Commun. Math. Phys. 262, 317–341 (2006)
Ehrhardt, T.: The asymptotics of a Bessel-kernel determinant which arises in Random Matrix Theory. Adv. Math. 225, 3088–3133 (2010)
Federbush, P.: A two-dimensional relativistic field theory. Phys. Rev. 121, 1247–1249 (1961)
Gamayun, O., Iorgov, N., Lisovyy, O.: How instanton combinatorics solves Painlevé VI, V and III’s. J. Phys. A 46, 335203 (2013)
Its, A., Lisovyy, O., Tykhyy, Y.: Connection Problem for the sine-gordon/painlevé III Tau-Function and Irregular Conformal Blocks. International Mathematics Research Notices, 22 pages (2014)
Its, A., Prokhorov, A.: Connection problem for the tau-function of the sine-gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach. International Mathematics Research Notice, 22 pages (2016)
Its, A., Prokhorov, A.: On some Hamiltonian properties of the isomonodromic tau functions. Rev. Math. Phys. 30(07), 1840008 (2018)
Its, A., Liovyy, O., Prokhorov, A.: Monodromy dependence and connection formulae for isomonodromic tau functions. Duke Math. J. 167(7), 1347–1432 (2018)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica D2, 306–352 (1981)
Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. RIMS, Kyoto Univ. 18, 1137–1161 (1982)
Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica 1D, 80–158 (1980)
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)
Lenard, A.: Some remarks on large Toeplitz determinants. Pacific J. Math 42, 137 (1972)
McCoy, B., Tracy, C., Wu, T.: Painlevé functions of the third kind. J. Math. Phys. 18, 1058–1092 (1977)
McCoy, B., Wu, T.: The Two-Dimensional Ising Model, 2nd edn. Dover Publications, USA (2014)
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov
Palmer, J.: Planar Ising Correlations Progress in Mathematical Physics 49. Birkäuser Boston, Inc, Boston (2007)
Ruijsenaars, S.: On the two-point functions of some integrable relativistic quantum field theories. J. Math. Phys. 24, 922 (1983)
Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum fields, III and IV. Publ. RIMS, Kyoto Univ. 15, 577 (1979). 15 (1979) 871
Tracy, C.: Asymptotics of a τ-function arising in the two-dimensional Ising model. Commun. Math. Phys. 142, 297–311 (1991)
Tracy, C., Widom, H.: Asymptotics of a class of solutions to the cylindrical Toda equations. Commun. Math. Phys. 190, 697–721 (1998)
Widom, H.: The strong szegö limit theorem for circular arcs. Indiana Univ. Math. J. 21, 277–283 (1971)
Widom, H.: On the solution of a Painlevé III equation. Math. Phys. Anal. Geom. 3, 375–384 (2000)
Wu, T.-T.: Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model, I. Phys. Rev. 149, 380–401 (1966)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The results of this article grew out of an eight week long Research Experience for Undergraduates (REU) program hosted at the University of Michigan in summer 2018. The work of T.B. is supported by the AMS and the Simons Foundation through a travel grant and W.W. acknowledges financial support provided by the Michigan Center for Applied and Interdisciplinary Mathematics. Both authors are grateful to C. Doering for stimulating discussions.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bothner, T., Warner, W. Short Distance Asymptotics for a Generalized Two-point Scaling Function in the Two-dimensional Ising Model. Math Phys Anal Geom 21, 37 (2018). https://doi.org/10.1007/s11040-018-9296-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-018-9296-y