Invasion percolation and the incipient infinite cluster in 2D

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] x [-n, n] is of order (n(2)pi(n),)(k), for k greater than or equal to 1, where pi(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(2)pi(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\ --> infinity. We also establish this when an invaded site v is chosen at random from a box of radius n, and n --> infinity.