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arXiv · 2610.26419

A connection inequality for the random-cluster model

Jianping Jiang·Sike Lang·Yuxuan Sun
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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.

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Jianping Jiang, Sike Lang, Yuxuan Sun. 2026-10-19. A connection inequality for the random-cluster model. https://arxiv.org/abs/2610.26419

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