This paper is studying the critical regime of the planar random-cluster model on \({\mathbb {Z}}^2\) with cluster-weight \(q\in [1,4)\). More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation (\(q = 1\)) and the FK-Ising model (\(q = 2\)). Finally, we prove new bounds on the one, two and four-arm exponents for \(q\in [1,2]\), as well as the one-arm exponent in the half-plane. These improve the previously known bounds, even for Bernoulli percolation.

本文正在研究具有簇权重\(q\in [1,4)\)的平面随机簇模型在\({\mathbb {Z}}^2\)上的临界状态。更准确地说，我们证明了四边形中的交叉估计，它们的边界条件是一致的，并且仅取决于它们的极值长度。它们特别暗示任何分形边界都被宏观簇所触及，其粗糙度或边界上的配置是一致的。此外，它们暗示原始集群和双集群之间的接口集合的任何子顺序扩展限制都是由非简单的循环构成的。我们还获得了一些所谓的 arm-events 的属性：三个通用临界指数（半平面中的两个臂，半平面中的三个臂和本体中的五个臂），准乘性和良好分离特性（即使臂不在原始和对偶之间交替），以及事实四臂指数严格小于 2。这些结果以前仅对伯努利渗透 ( \(q = 1\) ) 和 FK-Ising 模型 ( \(q = 2\) ) 已知。最后，我们证明了\(q\in [1,2]\)的一臂、二臂和四臂指数的新界限，以及半平面中的单臂指数。这些改进了先前已知的界限，即使对于伯努利渗透也是如此。