随机图上随机簇模型的尺度、相共存和算法
Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs
摘要：
对于$Δ\ge 5$和作为$Δ$的函数的大的$q$，我们给出了随机$Δ$ -正则图上随机团簇模型相变的详细图象.特别是，我们确定了临界状态下有序相和无序相的权重的极限分布，并证明了相关性的指数衰减和远离临界状态的中心极限定理。我们的技术是基于使用聚合物模型和集群扩展控制偏离有序和无序的基态。这些技术也产生有效的近似计数和采样算法的Potts和随机簇模型的随机$Δ$ -正则图在所有温度下，当$q$是大的。这包括临界温度，在该温度下，已知Glauber和Swendsen-Wang动力学的Potts模型混合缓慢。我们进一步证明了新的慢混合的结果马尔可夫链，最值得注意的是，Swendsen-Wang动态混合指数缓慢整个开区间包含的临界温度。这一点以前只在临界温度下才知道。我们的许多结果适用于更一般的$Δ$ -正则图满足小集扩张条件。
英文摘要：
For $Δ\ge 5$ and $q$ large as a function of $Δ$, we give a detailed picture of the phase transition of the random cluster model on random $Δ$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $Δ$-regular graphs at all temperatures when $q$ is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slow-mixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to $Δ$-regular graphs satisfying a small-set expansion condition.
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