平面无规团簇模型：临界相的分形性质
Planar random-cluster model: fractal properties of the critical phase
摘要：
本文研究了具有团权$q\in[1,4)$的$\mathbb Z^2$平面随机团模型的临界区。更准确地说，我们证明了四次是统一的边界条件，只依赖于他们的极值长度的交叉估计。它们特别意味着任何分形边界都被宏观团簇所触及，均匀地在其粗糙度或所述边界上的配置。此外，它们意味着原始和对偶集群之间的接口集合的任何子顺序缩放限制都是由非简单的循环组成的。我们还获得了一些所谓的臂事件的属性：三个普遍的临界指数（两个武器在半平面，三个武器在半平面和五个武器在散装），准乘法和良好的分离性能（即使当武器之间没有交替原始和对偶），以及四臂指数严格小于2的事实。这些结果以前只知道伯努利渗流（$q = 1$）和FK-Ising模型（$q = 2$）。最后，我们证明了$q\in[1,2]$的一，二和四臂指数的新的界限。这些改进了以前已知的边界，即使是伯努利渗流。
英文摘要：
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 said 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 arms exponents for $q\in[1,2]$. These improve the previously known bounds, even for Bernoulli percolation.
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