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

摘要

This paper presents three results on dependent site percolation on the square lattice. First, there exists no positively associated probability measure on ${0,1}^{mathbb {Z}^{2}}$ with the following properties: (a) a single infinite 0cluster exists almost surely, (b) at most one infinite 1∗cluster exists almost surely, (c) certain probabilities regarding 1∗clusters are bounded away from zero. Second, the coexistence of an infinite 1∗cluster and an infinite 0cluster has probability zero when the underlying probability measure is ergodic with respect to translations, positively associated, and satisfies the finite energy condition. The third result analyzes 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.