Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation

TL;DR

This paper investigates dependent site percolation on the square lattice, proving the impossibility of coexistence of infinite clusters under certain conditions and analyzing the structure of such clusters when positive association is absent.

Contribution

It establishes new non-coexistence results for infinite clusters in dependent percolation models and characterizes cluster structures without positive association.

Findings

No positively associated measure allows coexistence of infinite 0 and 1* clusters.

Ergodic, positively associated measures cannot have both infinite clusters coexist.

Infinite clusters imply infinitely many disjoint self-avoiding paths under finite energy conditions.

Abstract

This paper presents three results on dependent site percolation on the square lattice. First, there exists no positively associated probability measure on {0,1}^{Z^2} with the following properties: a) a single infinite 0cluster exists almost surely, b) at most one infinite 1*cluster exists almost surely, c) some probabilities regarding 1*clusters are bounded away from zero. Second, we show that coexistence of an infinite 1*cluster and an infinite 0cluster is almost surely impossible when the underlying probability measure is ergodic with respect to translations, positively associated, and satisfies the finite energy condition. The third result analyses the typical structure of infinite clusters of both types in the absence of positive association. Namely, under a slightly sharpened finite energy condition, the existence of infinitely many disjoint infinite self-avoiding 1*paths follows from the existence of an infinite 1*cluster. The same holds with respect to 0paths and 0clusters.