论文标题
在树独特性中随机常规图上的随机群集动力学
Random-cluster dynamics on random regular graphs in tree uniqueness
论文作者
论文摘要
我们将随机群集Glauber动态的快速混合在所有$ q \ ge 1 $和$ p <p_u（q，δ）$上的随机$δ$中，其中阈值$ p_u（q，Δ）$对应于（唯一/非独特/非独特相位）的随机模型（无单位）模型（infruster on Infruster）$Δ$Δ预计该阈值是尖锐的，对于$ q> 2 $，随机$δ$ -Recarbular图上的Glauber动力学会在$ p_u（q，δ）$上进行指数缩减。 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.作为推论，我们推断了swendsen的快速混合 - 在树唯一性区域中，每$ q \ ge 2 $的随机$δ$ - ge 2 $的wang动力学的快速混合。我们的证明依赖于“破碎时间”上的急剧绑定，即将任何配置分解为$ O（\ log n）$尺寸群集所需的步骤数。这是通过分析一种精致且新颖的迭代方案来确定的，以同时在给定时间揭示带有Glauber动力学配置簇的基本随机图。
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.
