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Painlevé Transcendents and Scaling Functions of the Two-Dimensional Ising Model

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Nonlinear Equations in Physics and Mathematics

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 40))

Abstract

In this lecture I should like to report on the work I have done in collaboration with Barry McCoy and Tai Tsun Wu (and on certain aspects with Eytan Barouch)on the correlation functions of the two-dimensional Ising model. In particular, I wish to demonstrate how a particular solution of the two-dimensional hyperbolic sine-Gordon equation (also known as the two-dimensional nonlinear Debye-Hückel equation),

$$\Delta\Phi=\,\sinh\Phi,\,\,(r\,>\,0),$$
(1.1)

plays a fundamental role in the scaled two-point function in both the one-phase and two-phase regions. Before I discuss these results, I would like first to review the definition of the 2-d. Ising model1)-4)and related quantities (Section 2); and then briefly recall the scaling theory hypothesis5)-7)for correlation functions (Section 3).

Lectures given at NATO Advanced Study Institute on Nonlinear Equations in Physics and Mathematics, Istanbul, August 1977.

Supported in part by National Science Foundation Grants Nos. PHY-76-15328 and PMR 73-07565 A01.

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References

References and Footnotes

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  18. B.M. McCoy, C.A. Tracy and T.T. Wu, unpublished notes.

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  19. The direct numerical evaluation of G±(2) (0) is done using the Painlevé transcendent introduced below.

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Note Added in Proof

  • For some new results, see B.M. McCoy and T.T. Wu, Phys. Lett. 72B, 219 (1977); R.Z. Bariev, 64A, 169 (1977); D. Wilkinson, to appear in Phys. Rev. D; and R. Haberman, Stud. Appl. Math. 57, 247 (1977).

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Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, State University of New York, Stony Brook, L.I., N.Y., 11794, USA

    Craig A. Tracy

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  1. Craig A. Tracy
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Editor information

Editors and Affiliations

  1. Department of Physics, University of Colorado, Boulder, Colo., USA

    A. O. Barut

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© 1978 D. Reidel Publishing Company, Dordrecht, Holland

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Tracy, C.A. (1978). Painlevé Transcendents and Scaling Functions of the Two-Dimensional Ising Model. In: Barut, A.O. (eds) Nonlinear Equations in Physics and Mathematics. NATO Advanced Study Institutes Series, vol 40. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9891-9_10

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  • DOI: https://doi.org/10.1007/978-94-009-9891-9_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-9893-3

  • Online ISBN: 978-94-009-9891-9

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