Resumen de The lowest crossing in two-dimensional critical percolation
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.
