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Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness
Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2021-04-17 , DOI: 10.1007/s00220-021-04093-z
Antonio Blanca , Reza Gheissari

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.



中文翻译:

树唯一性的随机正则图上的随机聚类动力学

我们在所有\(q \ ge 1 \)\(p <p_u(q,\ varDelta)\)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。

更新日期:2021-04-18
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