The lowest crossing in two-dimensional critical percolation

• 成果类型: Article
• 署名作者: Van den Berg, J; Járai, AA
• 署名单位: Centrum Wiskunde & Informatica (CWI); University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1241-1253
• 关键词: exponents

摘要

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 AB is of order (log(n/m))(-1), uniformly in 1 less than or equal to m less than or equal to n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.