Nonlinear differential equations for the correlation functions of the 2D Ising model on the cylinder

TL;DR

This paper derives nonlinear differential equations and determinant representations for the scaled 2-point functions of the 2D Ising model on a cylinder, extending classical results from infinite lattice cases.

Contribution

It introduces generalized nonlinear differential equations for the 2D Ising model's correlation functions on a cylinder, expanding the mathematical framework beyond infinite lattice scenarios.

Findings

Derived determinant representations for 2D Ising model correlations.

Established nonlinear differential equations generalizing Painlevé equations.

Extended classical results to cylindrical geometry.

Abstract

We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlev\'e III equation and the equation for the $\tau$-function of Painlev\'e V).