半平面非共存无需FKG
Half-plane non-coexistence without FKG
AI总结：
该研究证明在无正相关假设下，半平面渗透过程的非共存性，适用于多种随机模型。
摘要：
对于在无限正方形晶格上定义的边渗透测度μ，令μ_hp（分别，μ*_{hp}）表示其在上半平面的边缘（分别，其平面对偶过程的边缘）。我们证明，如果μ是平移不变、遍历且几乎必然只有有限个无限簇，则几乎必然μ_hp没有无限簇，或几乎必然μ*_{hp}没有无限簇。通过经典的Burton-Keane论证，这些假设如果μ是平移不变、遍历且具有有限能量，则成立。与之前的“非共存”定理不同，我们的结果不假定μ具有正相关（FKG）假设。我们的论证也适用于随机簇模型（包括q<1的区域，该区域缺乏FKG），均匀生成树和均匀奇子图。
英文摘要：
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.
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