On Isomonodromic Deformation of Massive Ising Spinors

TL;DR

This paper provides a rigorous derivation of the 2-point Ising spin correlation function in the massive scaling limit, connecting it to Painlevé transcendents and isomonodromic deformation theory, building on classical and recent discrete analysis results.

Contribution

It offers a concise, rigorous derivation of the continuous theory for Ising spin correlations using isomonodromic deformation, linking discrete and continuous frameworks.

Findings

Derived the 2-point correlation function in terms of Painlevé transcendent.

Connected discrete lattice results with continuous isomonodromic deformation theory.

Provided a rigorous account of the classical formula within modern analytical frameworks.

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 Painlev\'e 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.