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

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 $M\ne 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $M\ne 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\ne 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé 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 Painlevé 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 Painlevé 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 (Painlevé V) relations between the sum and difference of sigma's to this Painlevé VI framework.

arXiv comment46 pages

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