论文标题
$ν= \，-k $和m+n奇数，$ m \ le n $，$ t <t_c $及其lambda扩展
Factorization of Ising correlations C(M,N) for $ ν= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions
论文作者
论文摘要
我们研究了$ν= -k $和m+n奇数，$ m \ le n $的ising低温相关性C（m，n）的因素化。我们发现，$ M \ neq 0 $的两个因素满足相同的非线性微分方程，同样，对于M = 0，每个因素每个因素都满足pachlevéVI方程的冈本Sigma-Sigma-Sigma形式，并具有相同的Okamoto参数。使用Landen转换，我们以$ M \ neq 0 $的形式显示，实际上可以将以前的非线性微分方程减少为PainlevéVI方程的冈本Sigma形式。对于两个和四因素的情况，我们发现在PainlevéVI方程的冈本Sigma形式上存在一个一个参数家族，这将相关性C（M，N）的分解与相应的Sigma sigma sigma Sigma sigmoto Sigma sigma sigma sigma sigma sigma sigma sigma sequal-equare fore的添加分解的分解。在参数的特殊值中，C（m，n）因子的lambda扩展在第一和第二类的完整椭圆函数中减少到均匀的多项式。我们还概括了Sigma的总和与差异之间的一些Tracy-Widom（PainlevéV）与这个ParelevéVI框架之间的关系。
We study the factorizations of Ising low-temperature correlations C(M,N) for $ν=-k$ 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 Painlevé 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 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.
