论文标题
平面随机群集模型：临界相的分形特性
Planar random-cluster model: fractal properties of the critical phase
论文作者
论文摘要
本文正在研究$ \ Mathbb z^2 $上的Planar Random-Cluster模型的关键制度，并在[1,4）$中研究了cluster-weight $ q \。更确切地说，我们证明了四边形的交叉估计值，这些估计值在其边界条件下均匀，仅取决于它们的最大长度。他们特别暗示着任何分形边界都被宏观簇触摸，其粗糙度或上述边界的配置均匀地触及。此外，他们暗示着原始簇和双重簇之间接口收集的任何子顺序缩放限制都是由非简单的循环制成的。 我们还获得了许多所谓的手臂事件的特性：三个普遍的关键指数（半个平面中的两个武器，半平面中的三个武器和五个臂），批量的额外置换性和良好的分离属性（即使在原始和偶尔之间不交流的事实），以及以前较小的2臂指数，即使在武器之间交替交流。 （$ Q = 1 $）和FK-ing-asish型号（$ Q = 2 $）。 最后，我们在[1,2] $中以$ q \ $ q \ $ q \ $ q \ $ q \ in证明了新的界限。这些改善了以前已知的界限，即使是用于伯努利的渗透。
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.
