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$\Delta$-正则图上的有序/无序相变，对于所有足够大的$q$，并利用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，自然马尔可夫链（包括Glauber动力学）在随机正则图上的性能仍未得到充分理解，部分原因是该模型的非局域性（尤其在低温下），部分原因是在有序/无序相变周围窗口内出现的严重瓶颈现象。然而，人们普遍推测，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。我们的主要结果对所有足够大的$q$（相对于$\Delta$）建立了这一猜想。具体而言，我们考虑了从两种极端配置（全同和全异）初始化的Glauber动力学的混合时间，并得到了一对覆盖所有温度$\beta$的快速混合界，特别是包括瓶颈窗口在内。我们的结果受Gheissari和Sinclair近期对Ising模型方法的启发，他们为随机正则图在足够低温度下获得了类似风格的混合时间界。为了覆盖RC模型中的所有温度，我们适当地改进了Helmuth、Jenssen和Perkins关于有序/无序相变的结构性结果，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。