Factorization of percolation density correlation functions for clusters touching the sides of a rectangle
Abstract
We consider the density at a point z = x + i y of critical percolation clusters that touch the left [PL(z)], right [PR(z)], or both [PLR(z)] sides of a rectangular system, with open boundary conditions on the top and bottom. The ratio C(z) = PLR(z) / sqrt[PL(z) PR(z) Pih], where Pih is the probability of horizontal crossing given by Cardy, is a universal function of z and goes to a constant value C0 = 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 PL(z), PR(z), and PLR(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 C1 = 1.022.