The Ising correlation $C(M,N)$ for $\nu =-k$

Abstract

We present Painlev{é} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $C(M,N)$ with $M\le N$ in the special case $\nu =-k$ where $\nu =\sinh 2E_h/k_{B}T/\sinh 2E_v/k_{B}T$. More specifically four different non-linear ODEs depending explicitly on the two integers $M$ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $M+N$ even or odd. These four different non-linear ODEs are also valid for $M\ge N$ when $\nu =-1/k$. For the low-temperature row correlation functions $C(0,N)$ with $N$ odd, we exhibit again for this selected $\nu =-k$ condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this $\nu =-k$ case for $T<T_c$ and also $T>T_c$, that $C(M,N)$ with $M\le N$ is given as an $N\times N$ Toeplitz determinant.

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Cited by in corpus (3)

• Factorization of Ising correlations C(M,N) for $\nu =-k$ and M+N odd, $M\le N$, $T<T_c$ and their lambda extensions
• The lambda extensions of the Ising correlation functions C(M, N)
• On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model