Abstract:For $\mu$ an edge percolation measure on the infinite square lattice, let $\mu_{\textit{hp}}$ (respectively, $\mu^*_{hp}$) denote its marginal (respectively, the marginal of its planar dual process) on the upper half-plane. We show that if $\mu$ is translation-invariant and ergodic and almost surely has only finitely many infinite clusters, then either almost surely $\mu_{hp}$ has no infinite cluster, or almost surely $\mu^*_{hp}$ has no infinite cluster. By the classical Burton--Keane argument, these hypotheses are satisfied if $\mu$ 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 $\mu$. 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.
From: Frederik Ravn Klausen [view email]
[v1]
Thu, 12 Feb 2026 18:51:51 UTC (555 KB)
[v2]
Fri, 7 Aug 2026 07:04:41 UTC (1,179 KB)
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