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.

Small corrections (final version). In press, J. Phys. A

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• Short-range correlations in percolation at criticality
• Local logarithmic correlators as limits of Coulomb gas integrals
• Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus
• Critical points in coupled Potts models and correlated percolation
• On the analytical continuation of lattice Liouville theory
• Factorization Formulas for $2D$ Critical Percolation, Revisited
• Backbone three-point correlation function in the two-dimensional Potts model