The $q<1$ Random-Cluster Model on Wired Trees: Uniqueness and Negative Dependence

Abstract

The random-cluster model with cluster weight $0<q<1$ is expected to exhibit negative dependence, but without FKG even pairwise negative correlation remains open on general graphs. We study the model on the infinite $Δ$-regular tree with wired boundary conditions. Write $\widehat p:=p/(p+q(1-p))$ and $p_{\mathsf c}:=q/(Δ+q-2)$. Classical results identify the product wired state for $p\le p_{\mathsf c}$ and construct a percolative all-wired limit for $p>p_{\mathsf c}$. We prove that the supercritical wired DLR specification has a unique Gibbs measure, namely this all-wired limit. Consequently, the wired DLR phase diagram is complete: the unique measure is Bernoulli bond percolation with parameter $\widehat p$ for $p\le p_{\mathsf c}$, while it percolates for $p>p_{\mathsf c}$. We also establish negative dependence across wired branches. On a finite wired tree, the vector of branch-connectivity indicators satisfies conditional negative association under positive external fields (CNA+). Hence bounded increasing observables supported on disjoint collections of branches incident to a common vertex have nonpositive covariance. The same inequality holds in the unique infinite-volume wired measure for every $p$, with equality for $p\le p_{\mathsf c}$.

Disclosure

“ge sets. The argument requires a vertex whose incident branches separate the supports of the two observ- ables. Such a vertex need not exist for general disjoint supports. Disclosure of generative AI use OpenAI’s ChatGPT was used to assist with English-language editing, exposition, and pre- submission review of the manuscript. All mathematical ideas, arguments, and results are the author’s own, and the author takes full responsibility for the contents of t”

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2026-08-23T05:02:27.052922Z

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