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

TL;DR

This paper offers a simplified derivation of the constant factor in the short-distance asymptotics of the two-point function tau-function in the 2D Ising model, building on Tracy and Widom's previous work.

Contribution

It introduces a new derivation method based on an action integral representation and Painlevé-III asymptotics, simplifying prior complex analyses.

Findings

Derived the constant factor using a new approach

Connected tau-function asymptotics with Painlevé-III transcendent

Simplified previous derivations of the scaling constant

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 Painlev\'e-III transcendent from \cite{MTW}.