A short note on the scaling function constant problem in the two-dimensional Ising model

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 C. Tracy in \cite{T} via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom \cite{TW} 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 \cite{MTW}.

arXiv comment10 pages

References in corpus (4)

• Exact expectation values of local fields in quantum sine-Gordon model
• How instanton combinatorics solves Painlevé VI, V and III's
• Asymptotics of a Class of Solutions to the Cylindrical Toda Equations
• Connection problem for the tau-function of the Sine-Gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach

Cited by in corpus (2)

• Irregular conformal blocks and connection formulae for Painlevé V functions
• Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model