Half-plane non-coexistence without FKG

TL;DR

This paper proves a non-coexistence result for percolation models on the square lattice without relying on the FKG inequality, extending previous results to models lacking positive correlation.

Contribution

It establishes a non-coexistence theorem for infinite clusters in half-plane percolation models without the FKG condition, broadening applicability to models like the random-cluster model with q<1.

Findings

Either the half-plane marginal or its dual has no infinite cluster

The result applies to models without positive correlation (FKG)

The proof extends to uniform spanning trees and odd subgraphs

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.