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



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).


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

Mathematics Subject Classification

Primary 82B20 Secondary 70S05 34M55 


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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