Incipient infinite percolation clusters in 2D

Abstract

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC.

We also show that for any $k \ge 1$ the difference in size between the $k$th and $(k+1)$st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.