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 P3 at the percolation point in terms of the two-point density correlation functions P2. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z1, z2, z3) = P3(z1, z2, z3) [P2(z1, z2) P2(z1, z3) P2(z2, z3)]1/2 is not only universal but also a constant, independent of the zi, 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.
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