The Incipient Infinite Cluster Does Not Stochastically Dominate the Invasion Percolation Cluster in Two Di- Mensions

This note is motivated by results in [1, 8] about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions. In the classical mathematical theory of percolation, the edges (or vertices) of an infinite lattice are deleted independently with probability 1− p, and the properties of the remaining components are studied. There is a phase transition in the parameter: if p is bigger than some critical value pc , there is an infinite component with probability 1 (in this case we say that percolation occurs), and if p is below pc , the probability of the existence of an infinite component is 0. If p = pc , the geometric properties of connected components are highly non-trivial. It is expected (and has been proved for certain lattices, see, e.g., [17, 20]) that simple characteristics of connected components (e.g., diameter, volume) obey power laws. In other words, at criticality, connected components are self-similar random objects. Invasion percolation is a stochastic growth model [4, 18] that mirrors aspects of the critical percolation picture without tuning any parameter. To define the model, let G = (V, E) be an infinite connected graph in which a distinguished vertex, the origin, is chosen. The edges of G are assigned independent uniform random variables τe on [0, 1], called weights. (The underlying probability measure is denoted by P.) The invasion percolation cluster (IPC) of the origin on G is defined as the limit of an increasing sequence (Gn) of connected sub-graphs of G as follows. Define G0 to be the origin. Given Gn = (Vn, En), the edge set En+1 is obtained from En by adding to it the edge 1THE RESEARCH OF THE AUTHOR HAS BEEN SUPPORTED BY THE GRANT ERC-2009-ADG 245728-RWPERCRI.