Some two-dimensional finite energy percolation processes

TL;DR

This paper constructs translation invariant percolation processes on a 2D lattice that simultaneously exhibit infinite open and closed clusters, demonstrating coexistence under finite energy conditions.

Contribution

It introduces the first known example of a translation invariant percolation process with finite energy that shows coexistence of infinite open and closed clusters.

Findings

Existence of translation invariant percolation with infinite clusters of both types

Coexistence persists under independent thinning

Finite energy condition is compatible with coexistence

Abstract

Some examples of translation invariant site percolation processes on the $\Z^2$ lattice are constructed, the most far-reaching example being one that satisfies uniform finite energy (meaning that the probability that a site is open given the status of all others is bounded away from $0$ and $1$) and exhibits a.s. the coexistence of an infinite open cluster and an infinite closed cluster. Essentially the same example shows that coexistence is possible between an infinite open cluster and an infinite closed cluster that are both robust under i.i.d. thinning.