arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2008.02264math.PRcs.DMmath-phmath.MP
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树唯一性随机正则图上的随机簇动力学

Random-cluster dynamics on random regular graphs in tree uniqueness

Antonio Blanca, Reza Gheissari

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35 pages, 6 figures
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摘要：

我们建立快速混合的随机簇Glauber动力学随机$Δ$ -正则图的所有$q\ge 1$和$p<p_u(q,Δ)$，其中的阈值$p_u(q,Δ)$对应于一个唯一性/非唯一性相变的随机簇模型（无限）$Δ$ -正则树。预期该阈值是尖锐的，并且对于$q>2$，随机$Δ$ -正则图上的Glauber动力学在$p_u(q,Δ)$经历指数减速。更准确地说，我们表明，对于每一个$q\ge 1$，$Δ\ge 3$，$p<p_u(q,Δ)$，概率$1-o(1)$超过选择一个随机$Δ$ -正则图上的$n$顶点，格劳伯动力学的随机集群模型有$Θ(n \log n)$混合时间。作为推论，我们推导出了随机$Δ$ -正则图上Potts模型的Swendsen-Wang动力学的快速混合，对于每一个$q\ge 2$，在树的唯一性区域。我们的证明依赖于对“破碎时间”的严格限制，即，将任何配置分解为$O(\log n)$大小的群集所需的步骤数。这是通过分析一个微妙的和新颖的迭代方案，同时揭示了潜在的随机图与集群的Glauber动力学配置上，在给定的时间。

英文摘要：

We establish rapid mixing of the random-cluster Glauber dynamics on random $Δ$-regular graphs for all $q\ge 1$ and $p<p_u(q,Δ)$, where the threshold $p_u(q,Δ)$ corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $Δ$-regular tree. It is expected that this threshold is sharp, and for $q>2$ the Glauber dynamics on random $Δ$-regular graphs undergoes an exponential slowdown at $p_u(q,Δ)$. More precisely, we show that for every $q\ge 1$, $Δ\ge 3$, and $p<p_u(q,Δ)$, with probability $1-o(1)$ over the choice of a random $Δ$-regular graph on $n$ vertices, the Glauber dynamics for the random-cluster model has $Θ(n \log n)$ mixing time. As a corollary, we deduce fast mixing of the Swendsen--Wang dynamics for the Potts model on random $Δ$-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.

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