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

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

2007-04-06

arxiv.org/abs/0704.0901v2

J. Stat. Mech. (2007) P06012

Physics

Condensed Matter

Disordered Systems and Neural Networks

11 pages, 6 figures. Minor revisions

Scientific paper

10.1088/1742-5468/2007/06/P06012

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.

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