The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions

TL;DR

This paper demonstrates that in two-dimensional lattices, the incipient infinite cluster (IIC) does not stochastically dominate the invasion percolation cluster (IPC), highlighting a fundamental difference from the behavior observed on regular trees.

Contribution

The paper proves that on two-dimensional lattices, the IIC does not stochastically dominate the IPC, contrasting with known results on regular trees.

Findings

IIC does not stochastically dominate IPC in 2D lattices

The relation between IIC and IPC differs between trees and 2D lattices

First example showing this difference in stochastic domination

Abstract

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.