Factorization of correlations in two-dimensional percolation on the plane and torus

TL;DR

This paper confirms the factorization of three-point correlations in 2D percolation on large tori, explores its behavior on the torus, and provides a simplified expression relating it to SLE parameters, extending previous conformal invariance results.

Contribution

It numerically verifies the correlation factorization on the torus and analyzes its behavior, also deriving a simplified formula involving the SLE parameter kappa.

Findings

Confirmed factorization of correlations on large tori.

Identified a minimum value of the ratio R on the torus.

Derived a simplified expression for R as a function of kappa.

Abstract

Recently, Delfino and Viti have examined the factorization of the three-point density correlation function P_3 at the percolation point in terms of the two-point density correlation functions P_2. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z_1, z_2, z_3) = P_3(z_1, z_2, z_3) [P_2(z_1, z_2) P_2(z_1, z_3) P_2(z_2, z_3)]^{1/2} is not only universal but also a constant, independent of the z_i, and in fact an operator product expansion (OPE) coefficient. Delfino and Viti analytically calculate its value (1.022013...) for percolation, in agreement with the numerical value 1.022 found previously in a study of R on the conformally equivalent cylinder. In this paper we confirm the factorization on the plane numerically using periodic lattices (tori) of very large size, which locally approximate a plane. We also investigate the general behavior of R on the torus, and find a minimum value of R approx. 1.0132 when the three points are maximally separated. In addition, we present a simplified expression for R on the plane as a function of the SLE parameter kappa.