  This note is motivated by results in arXiv:math/0608132 and arXiv:0806.2425
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.
