论文标题
随机图上随机群集模型的有限尺寸缩放，相共存和算法
Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs
论文作者
论文摘要
对于$δ\ ge 5 $和$ q $ tamen的函数，我们详细介绍了随机$δ$ regratular图的随机群集模型的相变。特别是，我们确定在临界时有序和无序阶段的权重的限制分布，并证明相关性和中心限制定理的指数衰减远离关键性。 我们的技术基于使用聚合物模型和群集扩展来控制与有序和无序基础状态的偏差。这些技术还可以在$ q $大的所有温度下在所有温度下在随机$δ$中的POTTS和随机群集模型中产生有效的近似计数和采样算法。这包括在Potts模型中缓慢混合的Glauber和Swendsen-Wang动力学的临界温度。我们进一步证明了马尔可夫链的新慢速混合结果，最值得注意的是，在包含临界温度的开放时间间隔内，Swendsen-Wang Dynamics呈指数级成倍缓慢。以前仅在临界温度下已知。 我们的许多结果更普遍地适用于满足小型扩展条件的$δ$规范图。
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.
