Half-plane non-coexistence without FKG

TL;DR

This paper establishes half-plane non-coexistence for edge percolation on $\mathbb{Z}^2$ without the FKG hypothesis by proving a trichotomy for the half-plane marginals $\mu_{\mathrm{hp}}$ and $\mu^*_{\mathrm{hp}}$ under translation-invariance, ergodicity, and finite-energy with finitely many infinite clusters. The central strategy combines a Burton–Keane style uniqueness argument for the half-plane infinite cluster with planar duality, ruling out simultaneous infinite clusters in the primal and dual halves. The results extend to the random-cluster model (including the $q<1$ regime lacking FKG), the uniform spanning tree, and the uniform odd subgraph, yielding half-plane non-coexistence and weak non-coexistence statements near the self-dual point. The work broadens non-coexistence phenomena beyond positive association, offering robust methods for analyzing phase behavior in the half-plane across several planar systems and suggesting several open directions for full-plane extensions and related models.

Abstract

For $μ$ an edge percolation measure on the infinite square lattice, let $μ_{\textit{hp}}$ (respectively, $μ^*_{hp}$) denote its marginal (respectively, the marginal of its planar dual process) on the upper half-plane. We show that if $μ$ is translation-invariant and ergodic and almost surely has only finitely many infinite clusters, then either almost surely $μ_{hp}$ has no infinite cluster, or almost surely $μ^*_{hp}$ has no infinite cluster. By the classical Burton--Keane argument, these hypotheses are satisfied if $μ$ is translation-invariant and ergodic and has finite-energy. In contrast to previous ``non-coexistence'' theorems, our result does not impose a positive-correlation (FKG) hypothesis on $μ$. Our arguments also apply to the random-cluster model (including the regime $q<1$, which lacks FKG), the uniform spanning tree, and the uniform odd subgraph.

Figures (5)

• Figure 1: If three infinite clusters (solid lines) in the half-plane touch the box, then they are separated by two dual clusters (dotted lines). The middle cluster cannot glue to any other cluster when we reveal the lower-half-plane.
• Figure 2: Removing a finite-height vertical strip can change the number of infinite clusters. The first two examples show non-tenuous infinite clusters, and the third shows a tenuous infinite cluster.
• Figure 3: The solid lines connect the positive and negative $x$-axes to the unique infinite primal cluster. This creates a finite region which traps the dual cluster containing the origin plaquette.
• Figure 4: The open (solid edges) and complement (dashed edges) can "cross" only if they are arranged as on the left, where the central vertex has degree $2$. There can be no such "crossings" in an odd subgraph, since each vertex is as depicted on the right.
• Figure 5: One step in the exploration of the cluster containing the vertex $(0,n-1)$. The blue region has been previously revealed, and the yellow region is still unexplored. If all of the orange edges are closed, then the cluster gets "cut off" and must be finite.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4