论文标题
空间混合和晶格上的随机群集动力学
Spatial mixing and the random-cluster dynamics on lattices
论文作者
论文摘要
在理解旋转系统Glauber动力学混合时间的重要范式中，模型的空间混合属性与在动力学混合时间上的边界之间的对应关系。这尤其包括弱和强空间混合的经典概念，这些概念被用来显示Ising和Potts模型的Glauber动力学的高温态度中最著名的混合时间边界。 随机群集模型的Glauber动力学自然不适合此自旋系统框架，因为其过渡规则不是局部的。在本文中，我们介绍了弱空间混合，强空间混合与阶段内空间混合的较新概念之间的各种含义，以及在$ \ mathbb z^d $中的随机群集动力学的混合时间范围，用于$ \ mathbb z^d $ for $ d \ ge 2 $。这些暗示了许多新结果，包括最佳$ O（N \ log n）$混合Torii上的随机聚类动态和$ n $ tho $ n $顶点的盒子在所有高温下，并且在$ q $ q $ q $ quasi-polynomial（或quasi-polynomial on Tori on Tori on Tori in y时）的$ \ Mathbb z^d $ n时（或quasi-quasi-quasi on n $ quasi-quasi-quasiiz oon Toriiz ins nsiz nsiz in y时）在临界点（相比之下，最坏情况初始化的混合时间呈指数较大）。在相同的参数制度中，这些结果转化为Potts模型的快速采样算法，$ \ Mathbb z^d $ for General $ d $。
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random-cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this paper, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random-cluster dynamics in finite subsets of $\mathbb Z^d$ for general $d\ge 2$. These imply a host of new results, including optimal $O(N\log N)$ mixing for the random cluster dynamics on torii and boxes on $N$ vertices in $\mathbb Z^d$ at all high temperatures and at sufficiently low temperatures, and for large values of $q$ quasi-polynomial (or quasi-linear when $d=2$) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst-case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on $\mathbb Z^d$ for general $d$.
