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 $(\sin \pi y{)}^{-5/48}\{[\cos (\pi y/2){]}^{1/3}+[\sin (\pi y/2){]}^{1/3}-1\}$. 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.

arXiv comment11 pages, 6 figures. Minor revisions

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• Critical p=1/2 in percolation on semi-infinite strips