Invasion Percolation and the Incipient Infinite Cluster in 2D

Abstract

We establish two links between two-dimensional invasion percolation and Kesten's incipient infinite cluster (IIC). We first prove that the k th moment of the number of invaded sites within the box [−n, n]×[−n, n] is of order (n 2π n ) k , for k≥1, where π n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n. This improves a result of Y. Zhang. We show that the size of the invaded region, when scaled by n 2π n , is tight. Secondly, we prove that the invasion cluster looks asymptotically like the IIC, when viewed from an invaded site v, in the limit |v|→∞. We also establish this when an invaded site v is chosen at random from a box of radius n, and n→∞.