Percolação orientada crítica em duas dimensões

Abstract

We’ll investigate the critical oriented percolation process in Z2 through results similar to the Russo-Seymour-Welsh. The objective of this work is to detail the arguments made in The box-crossing property for critical two-dimensional oriented percolation [3] to bound connection properties of the critical model. Primarily we’ll prove results similar to the technology of Russo-Seymour-Welsh for oriented percolation in Z2. Then we’ll find box proportions that guarantee crossings with non-trivial probabilities. Finally, we’ll use the theorems shown to bound from above and below the probability that the cluster of the origin extends beyond distance n and the typical width of the cluster on height n, conditioned on the event that the cluster has reached this level.