矩形边界上团簇渗流密度关联函数的因式分解
Factorization of percolation density correlation functions for clusters touching the sides of a rectangle
摘要：
本文研究了在上、下开边界条件下，接触矩形系统左[P_L（z）]、右[P_R（z）]或两侧[ P_{LR}（z）]的临界渗流团簇在z = x + y点处的密度.比率C（z）= P_{LR}（z）/ sqrt[P_L（z）P_R（z）Pi_h]，其中Pi_h是由Cardy给出的水平交叉的概率，是z的通用函数，并且达到恒定值C_0 = 2^（7/2）3^（-3/4）pi^（5/2）Gamma（1/3）^（-9/2）= 1.0299268.远离终点。我们在数值上观察到，对于有线b.c.，C（z）取决于x而不是y，这个结果通过共形场论导出了C（z）的显式表达式。对于半无限带，我们也得到了显式表达式的P_L（z），P_R（z），和P_{LR}（z），有线和开放的b.c.我们的结果使计算有限大小的修正的因式分解附近的一个孤立的锚点，锚定在两个边界点的集群的情况下。最后，我们给出了在水平方向上具有周期b.c.的矩形的数值结果，发现C（z）趋近于常数C_1 = 1.022。
英文摘要：
We consider the density at a point z = x + i y of critical percolation clusters that touch the left [P_L(z)], right [P_R(z)], or both [P_{LR}(z)] sides of a rectangular system, with open boundary conditions on the top and bottom. The ratio C(z) = P_{LR}(z) / sqrt[P_L(z) P_R(z) Pi_h], where Pi_h is the probability of horizontal crossing given by Cardy, is a universal function of z and goes to a constant value C_0 = 2^(7/2) 3^(-3/4) pi^(5/2) Gamma(1/3)^(-9/2) = 1.0299268... far from the ends. We observe numerically that C(z) depends upon x but not y for wired b.c., and this result leads to an explicit expression for C(z) via conformal field theory. For the semi-infinite strip we also derive explict expressions for P_L(z), P_R(z), and P_{LR}(z), for both wired and open b.c. Our results enable calculation of the finite-size corrections to the factorization near an isolated anchor point, for the case of clusters anchored at two boundary points. Finally, we present numerical results for a rectangle with periodic b.c. in the horizontal direction, and find that C(z) approaches a constant value C_1 = 1.022.
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