We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

中文翻译：

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通过决策树实现随机集群和 Potts 模型的急剧相变

我们证明了单调测度的决策树上的不等式，它概括了乘积空间上的OSSS不等式。作为一个应用，我们使用这个不等式来证明晶格自旋模型及其随机簇表示的许多新结果。更准确地说，我们证明了 1. 对于传递图上的 Potts 模型，对于 $\beta<\beta_c$，相关性以指数方式快速衰减。2. 对于传递图上簇权重为 $q\geq1$ 的随机簇模型，相关性在亚临界状态下呈指数衰减，并且簇密度在超临界状态下满足平均场下界。3. 对于平面拟传递图上簇权重为 $q\geq1$ 的随机簇模型 $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G} ^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ 作为特例，我们获得了正方形、三角形和六边形晶格的临界点值（这为 Beffara 和 Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]）。这些结果对于理解所有这些模型的亚临界（分别为无序）相有许多应用。本文开发的技术有可能扩展到广泛的模型类别，包括 Ashkin-Teller 模型、连续渗流模型（例如 Voronoi 渗流和布尔渗流）、大规模高斯自由场的超级集以及随机聚类和具有无限范围相互作用的 Potts 模型。三角形和六边形晶格（这提供了 Beffara 和 Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012] 结果的简短证明）。这些结果对于理解所有这些模型的亚临界（分别为无序）相有许多应用。本文开发的技术有可能扩展到广泛的模型类别，包括 Ashkin-Teller 模型、连续渗流模型（如 Voronoi 渗流和布尔渗流）、大规模高斯自由场的超级集以及随机聚类和具有无限范围相互作用的 Potts 模型。三角形和六边形晶格（这提供了 Beffara 和 Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012] 结果的简短证明）。这些结果对于理解所有这些模型的亚临界（分别为无序）相有许多应用。本文开发的技术有可能扩展到广泛的模型类别，包括 Ashkin-Teller 模型、连续渗流模型（如 Voronoi 渗流和布尔渗流）、大规模高斯自由场的超级集以及随机聚类和具有无限范围相互作用的 Potts 模型。