We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.

中文翻译：

---

随机聚类模型的信息渗滤与截止

我们认为，随机聚类模型（RCM）上具有参数p ∈（0,1）和q  ≥1.这是标准的键渗滤的概括（带边缘与概率独立地打开p），其由一个因子偏置q提高到已连接组件的数量。我们在此模型上研究了著名的Fortuin-Kasteleyn（FK）动力学，其中边沿的更新取决于系统的整体几何形状，这与旋转系统的Glauber热浴动力学不同，并证明了对于所有足够小的p（取决于尺寸），以及任何q > 1，FK-动力学显示出截止现象在具有窗口大小，其中λ _∞是该过程的光谱间隙的大n极限。我们的证明将Lubetzky和Sly的信息渗透框架扩展到了RCM，并且还依赖于Blanca和Sinclair的论证，他们证明了平面版本的混合时间很短。我们证明的一个关键方面是分析从动力学的图形构造中提取的一系列相关的（跨时间的）伯努利渗滤对信息传播的影响。