Factorization of percolation density correlation functions for clusters touching the sides of a rectangle

Understand

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.