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

TL;DR

This paper derives and verifies universal formulas for the density correlation functions of critical percolation clusters touching sides of a rectangle, using conformal field theory and numerical simulations.

Contribution

It provides explicit conformal field theory-based expressions for cluster densities and their ratios in rectangular geometries with various boundary conditions, including finite-size corrections.

Findings

C(z) approaches 1.03 far from edges in open boundary conditions.

C(z) approaches 1.022 in periodic boundary conditions.

Explicit formulas for cluster densities in semi-infinite strips.

Abstract

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.