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 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.
Keywords
Cite
@article{arxiv.1011.1101,
title = {Factorization of correlations in two-dimensional percolation on the plane and torus},
author = {Robert M. Ziff and Jacob J. H. Simmons and Peter Kleban},
journal= {arXiv preprint arXiv:1011.1101},
year = {2015}
}
Comments
Small corrections (final version). In press, J. Phys. A