Abstract
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 $R_n$ from the half-line left of A to the half-line right of B. We show that the probability that $R_n$ has a site at distance smaller than m from $\mathit{AB}$ is of order $(\log (n/m))^{-1}$, uniformly in $1 \leq m \leq n/2$. Much of our analysis can be carried out for other two-dimensional lattices as well.
Citation
A. A. Járai. J. van den Berg. "The lowest crossing in two-dimensional critical percolation." Ann. Probab. 31 (3) 1241 - 1253, July 2003. https://doi.org/10.1214/aop/1055425778
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