The density of critical percolation clusters touching the boundaries of strips and squares

Abstract

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to (sinπy)−5/48{[cos(πy/2)]1/3+[sin(πy/2)]1/3−1}. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides and compare with theory for the density along an edge.