The lowest crossing in 2D critical percolation

Details The lowest crossing in 2D critical percolation The lowest crossing in 2D critical percolation

2002-01-04

arxiv.org/abs/math/0201030v1

Ann. Probab. 31 (2003), no. 3, 1241-1253

Mathematics

Probability

16 pages, Latex, 2 eps figures, special macros: percmac.tex. Submitted to Annals of Probability

Scientific paper

10.1214/aop/1055425778

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.

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