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 $\Delta$-regular tree with wired boundary conditions. Write $\hat{p}:=p/(p+q(1-p))$ and $p_{\mathsf{c}}:=q/(\Delta +q-2)$. 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 $\hat{p}$ 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 $p\le p_{\mathsf{c}}$.

32 pages, 1 figure