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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).

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Notes

  1. Without using the Hamiltonian system (1.10) for the radial sinh-Gordon equation, but instead polynomial Hamiltonians for Painlevé-III, Okamoto [13, (7) and (8)], has used a similar approach in the identification of \(\tau _{\pm }(t,\lambda )\) as a tau-function product.

  2. It is in fact already present in [9] where \(A\left( \frac{1}{\pi }\right) \) is computed from a field theoretic viewpoint.

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  1. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Thomas Bothner

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  1. Thomas Bothner
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Correspondence to Thomas Bothner.

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The author is grateful to C. Tracy and A. Its for stimulating discussions about this project. This work is supported by the AMS and the Simons Foundation through a travel grant.

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Bothner, T. A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model. J Stat Phys 170, 672–683 (2018). https://doi.org/10.1007/s10955-017-1947-z

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