We establish rapid mixing of the random-cluster Glauber dynamics on random \(\varDelta \)-regular graphs for all \(q\ge 1\) and \(p<p_u(q,\varDelta )\), where the threshold \(p_u(q,\varDelta )\) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) \(\varDelta \)-regular tree. It is expected that this threshold is sharp, and for \(q>2\) the Glauber dynamics on random \(\varDelta \)-regular graphs undergoes an exponential slowdown at \(p_u(q,\varDelta )\). More precisely, we show that for every \(q\ge 1\), \(\varDelta \ge 3\), and \(p<p_u(q,\varDelta )\), with probability \(1-o(1)\) over the choice of a random \(\varDelta \)-regular graph on n vertices, the Glauber dynamics for the random-cluster model has \(\varTheta (n \log n)\) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random \(\varDelta \)-regular graphs for every \(q\ge 2\), in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into \(O(\log n)\) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

我们在所有\（q \ ge 1 \）和\（p <p_u（q，\ varDelta）\）的随机\（\ varDelta \）正则图上建立随机集群Glauber动力学的快速混合，其中阈值\（p_u（q，\ varDelta）\）对应于（无限）\（\ varDelta \）规则树上随机集群模型的唯一性/非唯一性相变。预期该阈值是尖锐的，并且对于\（q> 2 \），随机\（\ varDelta \） -正则图上的Glauber动力学在\（p_u（q，\ varDelta）\）处经历指数下降。更确切地说，我们表明对于每个\（q \ ge 1 \），\（\ varDelta \ ge 3 \），和\（p <p_u（q，\ varDelta）\），在n个顶点上选择随机\（\ varDelta \）正则图的概率为（{o-o（1）\），Glauber动力学对于随机集群模型，其混合时间为\（\ varTheta（n \ log n）\）。作为推论，我们推导出在树的唯一性区域中每个\（q \ ge 2 \）的随机\（\ varDelta \） -正则图上Potts模型的Swendsen-Wang动力学快速混合。我们的证明依赖于“中断时间”的严格界限，即将任何配置分解为\（O（\ log n）\）所需的步骤数大小的集群。这是通过分析精致且新颖的迭代方案来建立的，以在给定时间同时显示其上具有Glauber动力学配置簇的基础随机图。