Half-plane non-coexistence without FKG

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.

💡 Research Summary

The paper studies edge‑percolation measures μ on the infinite square lattice ℤ² and investigates the coexistence of infinite clusters in the upper half‑plane. For a given μ we denote by μ_hp (resp. μ*_hp) the marginal of μ (resp. of its planar dual μ*) restricted to the half‑plane ℤ×ℤ≥0. The classical non‑coexistence results of Zhang and Burton–Keane show that, under translation‑invariance, finite‑energy, and positive association (FKG), the full‑plane percolation and its dual cannot both contain infinite clusters. The novelty of this work is that the authors remove the FKG hypothesis entirely. They prove that translation‑invariance, ergodicity, and the condition that μ almost surely has only finitely many infinite clusters are sufficient to guarantee a “half‑plane non‑coexistence” phenomenon: either μ_hp has a unique infinite cluster and μ*_hp has none, or vice‑versa, or both have no infinite clusters.

The main technical result (Theorem 1.2) states that for any translation‑invariant, ergodic edge‑percolation measure μ on ℤ² with almost surely finitely many infinite clusters, the trichotomy above holds. The proof proceeds in two major steps. First, the authors show that μ_hp can have at most one infinite cluster. This is achieved by combining a planar‑topology argument with a Burton–Keane‑type bound on the number of “ends” of an infinite cluster. Lemma 2.2 establishes that each infinite cluster must contain vertices with arbitrarily large positive and negative y‑coordinates; Lemma 2.3 then uses this to rule out the possibility of two or more disjoint infinite clusters in the half‑plane, because such a configuration would force an infinite‑plane cluster to have three ends, contradicting the known bound that any translation‑invariant percolation on an amenable graph has at most two ends per infinite cluster.

The second step eliminates the possibility that both μ_hp and μ*_hp contain infinite clusters simultaneously. Assuming μ_hp has a unique infinite cluster, the authors prove that the plaquette centred at (½,½) cannot be connected to infinity in the dual configuration. The argument relies on the fact that the unique primal infinite cluster must intersect the positive and negative x‑axes arbitrarily far out, thereby “blocking’’ any dual path from the origin to infinity. A crucial intermediate result (Proposition 2.4) shows that under the same hypotheses there are no “tenuous’’ infinite clusters—clusters that become finite when restricted to a sufficiently high horizontal strip. This eliminates pathological cases where the number of infinite clusters could fluctuate under vertical translations, ensuring that the number of infinite clusters in μ_hp is almost surely constant (0 or 1).

The authors then apply the general theorem to several important models that lack the FKG property. For Bernoulli(p=½) percolation, Corollary 1.3 (via finite‑energy) yields a new proof that the half‑plane contains no infinite cluster without invoking Harris’s inequality. For the random‑cluster model, the result holds for all q>0, including the previously inaccessible regime q<1 where FKG fails. The uniform spanning tree (UST) on ℤ² is self‑dual and ergodic, yet it does not satisfy finite‑energy; nevertheless, Corollary 1.3 shows that its half‑plane marginal almost surely has only finite clusters. Finally, the uniform odd subgraph, which is self‑complementary rather than self‑dual, also satisfies the half‑plane non‑coexistence property by an adaptation of the duality argument.

The paper also discusses the necessity of the three hypotheses. Translation‑invariance alone is insufficient: the dual of the UST provides a counterexample where both primal and dual have infinite clusters. The authors construct a translation‑invariant, finite‑energy edge‑percolation measure with exactly two infinite clusters, demonstrating that the “finitely many infinite clusters’’ condition cannot be dropped. They note that the results extend to any half‑plane bounded by a line of rational slope and to other planar graphs admitting a ℤ²‑translation action with finitely many orbits.

In summary, the work establishes a robust half‑plane non‑coexistence theorem that does not rely on positive correlation. It broadens the class of planar percolation models for which one can guarantee that at most one of the primal or dual processes percolates in a half‑plane, and it provides new insights into the structure of infinite clusters under minimal symmetry and ergodicity assumptions.