A connection inequality for the random-cluster model

Source abstract

For a finite graph $G=(V,E)$, let $φ$ be the random-cluster measure with edge parameters $(p_e)_{e\in E}$, cluster weight $q\geq 1$ and free boundary condition. For each $x\in V$ and all nonempty $A,B\subseteq V$, we prove $φ(x\leftrightarrow B)\geq φ(x\leftrightarrow A)\min_{a\in A}φ(a\leftrightarrow B)$. For $q=1$, this is the (slightly generalized) Kozma-Nitzan inequality. Our extension to $q>1$ is based on a recent proof of this inequality by Blanc-Renaudie. As an application, we give a simple proof of sharp upper bounds for one-arm and point-to-plane connection probabilities in the subcritical regime.