Factorization of Ising correlations C(M,N) for $ \nu= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions

TL;DR

This paper explores the factorization of Ising model correlations at low temperature for specific parameter regimes, revealing connections to Painlevé VI equations and introducing lambda extensions that generalize correlation factorizations.

Contribution

It demonstrates that Ising correlations can be factorized into solutions of Painlevé VI equations with lambda extensions, unifying different cases and extending previous results.

Findings

Correlations factorize into Painlevé VI solutions with lambda parameters.

Landen transformation reduces nonlinear equations to Painlevé VI form.

Lambda extensions generalize correlation factorizations and connect to elliptic functions.

Abstract

We study the factorizations of Ising low-temperature correlations C(M,N) for $\nu=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlev\'e VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlev\'e VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlev\'e VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlev\'e VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlev\'e V) relations between the sum and difference of sigma's to this Painlev\'e VI framework.