The lowest crossing in two-dimensional critical percolation

• A. A. Járai ^[2] ; J. van den Berg ^[1]
  1. [1] University of British Columbia

     University of British Columbia

     Canadá

     
  2. [2] CWI, Amsterdam
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 3, 2003, págs. 1241-1253
• Idioma: inglés
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• Resumen

  • 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 Rn from the half-line left of A to the half-line right of B. We show that the probability that Rn 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.