We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $\beta>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $\Delta$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $\Delta$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $\beta$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties ''within the phase'', which are then related to the evolution of the chain.

翻译：我们考虑具有实参数$q>1$和温度$\beta>0$的随机团簇模型中Glauber动力学的性能。Helmuth、Jenssen和Perkins近期的工作详细描述了该模型在随机$\Delta$-正则图上对于所有充分大$q$的有序/无序转变，并利用团簇展开方法获得了对所有温度$\beta$的高效采样算法。尽管取得了这一重大进展，但自然马尔可夫链（包括Glauber动力学）在随机正则图上的性能尚未被充分理解，部分原因在于模型的非局域性质（尤其在低温下），以及围绕有序/无序转变的窗口内出现的严重瓶颈现象。然而，普遍猜想表明，通过更明智地初始化链，可以避免从最坏情况初始配置中阻碍混合的瓶颈现象。我们的主要结果针对所有充分大$q$（相对于$\Delta$）验证了这一猜想。具体而言，我们考虑从两种极端配置（全内和全外）初始化的Glauber动力学的混合时间，并得到一对覆盖所有温度$\beta$（特别是包括瓶颈窗口）的快速混合界。我们的结果受Gheissari和Sinclair近期对伊辛模型方法的启发，他们在随机正则图上对充分低温度获得了类似风格的混合时间界。为覆盖随机团簇模型中的所有温度，我们适当细化了Helmuth、Jenssen和Perkins关于有序/无序转变的结构性结果，并展示了“相内”的空间混合性质，这些性质随后与链的演化相关联。