Open Access
July 2003 The lowest crossing in two-dimensional critical percolation
A. A. Járai, J. van den Berg
Ann. Probab. 31(3): 1241-1253 (July 2003). DOI: 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 $R_n$ 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.


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A. A. Járai. J. van den Berg. "The lowest crossing in two-dimensional critical percolation." Ann. Probab. 31 (3) 1241 - 1253, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1087.60076
MathSciNet: MR1988471
Digital Object Identifier: 10.1214/aop/1055425778

Primary: 60K35

Keywords: critical exponent. , Critical percolation , lowest crossing

Rights: Copyright © 2003 Institute of Mathematical Statistics

Vol.31 • No. 3 • July 2003
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