Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model

TL;DR

This paper refines techniques to solve a generalized connection problem related to the two-point scaling function in the 2D Ising model, extending the understanding of its asymptotic behavior and relation to Painlevé functions.

Contribution

It provides a solution to a generalized connection problem for the two-point scaling function, broadening the scope of previous results in the Ising model.

Findings

Solution to a generalized connection problem

Refinement of techniques for asymptotic analysis

Enhanced understanding of Painlevé functions in Ising model

Abstract

In the 1977 paper \cite{MTW} of B. McCoy, C. Tracy and T. Wu it was shown that the limiting two-point correlation function in the two-dimensional Ising model is related to a second order nonlinear Painlev\'e function. This result identified the scaling function as a tau-function and the corresponding connection problem was solved by C. Tracy in 1991 \cite{T}, see also the works by C. Tracy and H. Widom in 1998 \cite{TW}. Here we present the solution to a certain generalized version of the above connection problem which is obtained through a refinement of the techniques in \cite{B}.