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

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 z i and in fact an operator product expansion coefficient. Delfino and Viti analytically calculated its value (1.022 013...) 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 ≈ 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 _.