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A connection inequality for the random-cluster model
Abstract
For a finite graph G=(V,E), let ϕ be the random-cluster measure with edge parameters (pe)e∈ E, cluster weight q≥ 1 and free boundary condition. For each x∈ V and all nonempty A,B⊆ V, we prove ϕ(x↔ B)≥ ϕ(x↔ A)mina∈ Aϕ(a↔ 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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BibTeX
@misc{jiang2026connection,
title = {{A connection inequality for the random-cluster model}},
author = {Jianping Jiang and Sike Lang and Yuxuan Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26419},
eprint = {2610.26419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26419},
}
RIS
TY - UNPB
TI - A connection inequality for the random-cluster model
AU - Jianping Jiang
AU - Sike Lang
AU - Yuxuan Sun
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26419
UR - https://arxiv.org/abs/2610.26419
AB - For a finite graph G=(V,E), let ϕ be the random-cluster measure with edge parameters (p_e)_e∈ E, cluster weight q≥ 1 and free boundary condition. For each x∈ V and all nonempty A,B⊆ V, we prove ϕ(x↔ B)≥ ϕ(x↔ A)min_a∈ Aϕ(a↔ 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.
ER -