On Isomonodromic Deformation of Massive Ising Spinors

Abstract

In this short note, we give a self-contained derivation of the formula for the $2$-point full-plane Ising spin correlation function under massive scaling limit in terms of a third Painlevé transcendant. This formula, first derived in a celebrated work of Wu, McCoy, Tracy, and Barouch, was subsequently reformulated in terms of the theory of isomonodromic deformation by Sato, Miwa, and Jimbo. In view of recent developments in the discrete analysis which have enabled, in particular, a convergence proof of spin correlation functions on isoradial lattice, we give a concise and rigorous account of the continuous theory in the same framework yielding this iconic result.

15 pages