Factorization of percolation density correlation functions for clusters touching the sides of arectangle

Abstract

In this paper we consider the density, at a pointz = x+iy, of critical percolation clusters that touch the left(PL(z)), right(PR(z)), orboth (PLR(z)) sides of a rectangular system, with open boundary conditions on the top and bottomsides. While each of these quantities is non-universal and indeed vanishes in the continuumlimit, the ratio , where Πh is the probability of left–right crossing given by Cardy, is a universal function ofz. With wired (fixed) boundary conditions on the left- and right-hand sides,high-precision numerical simulations and theoretical arguments show thatC(z) goes to aconstant C0 = 27/2 3−3/4 π5/2 Γ(1/3)−9/2 = 1.029 9268... for points far from the ends, and varies by no more than a few per cent for allz values.Thus PLR(z) factorizes over the entire rectangle to very good approximation. In addition, the numerical observation thatC(z) depends uponx but not upony leads to an explicitexpression for C(z) via conformal field theory for a long rectangle (semi-infinite strip). We also derive explicit expressionsfor PL(z), PR(z), andPLR(z) in this geometry,first by assuming y independence and then by a full analysis that obtains these quantities exactly with no assumption onthe y behavior. In this geometry we obtain, in addition, the corresponding quantitiesin the case of open boundary conditions, which allows us to calculateC(z) in the open system. We give some theoretical results for an arbitrary rectangle as well. Ourresults also enable calculation of the finite-size corrections to the factorization near anisolated anchor point, for the case of clusters anchored at points. Finally, we presentnumerical results for a rectangle with periodic b.c. in the horizontal direction, and findC(z) that approachesa constant value C1≈1.022.