We consider the problem of sampling from the ferromagnetic $q$-state Potts model on the random $d$-regular graph with parameter $β>0$. A key difficulty that arises in sampling from the model is the existence of a metastability window $(β_u,β_u')$ where the distribution has two competing modes, the so-called disordered and ordered phases, causing MCMC-based algorithms to be slow mixing from worst-case initialisations. To this end, Helmuth, Jenssen and Perkins designed a sampling algorithm that works for all $β$ when $q$ is large, using cluster expansion methods; more recently, their analysis technique has been adapted to show that random-cluster dynamics mixes fast when initialised more judiciously. However, a bottleneck behind cluster-expansion arguments is that they inherently only work for large $q$, whereas it is widely conjectured that sampling is possible for all $q,d\geq 3$. The only result so far that applies to general $q,d\geq 3$ is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast for $β<β_u$. For $β>β_u$, certain correlation phenomena emerge because of the metastability which have been hard to handle, especially for small $q$ and $d$. Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold $β_u$. We use planting as the main tool in our proofs, and combine it with the analysis of random-cluster dynamics. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers $q,d\geq 3$ beyond the uniqueness threshold $β_u$; our analysis works all the way up to the threshold $β_c\in (β_u,β_u')$ where the dominant mode switches from disordered to ordered. We also obtain an algorithm in the ordered regime $β>β_c$ that refines significantly the range of $q,d$.

翻译：我们研究在参数$β>0$下，从随机$d$-正则图上的铁磁$q$-态Potts模型中采样的问题。采样过程中的一个关键难点在于存在一个亚稳态窗口$(β_u,β_u')$，其中分布具有两个相互竞争的模态，即所谓的无序相和有序相，这导致基于MCMC的算法在最坏情况初始化下混合缓慢。为此，Helmuth、Jenssen和Perkins设计了一种在$q$较大时对所有$β$有效的采样算法，该算法使用了团簇展开方法；最近，他们的分析技术已被调整用于证明随机团簇动力学在更审慎的初始化下能够快速混合。然而，团簇展开论证的一个固有瓶颈是它们本质上仅适用于较大的$q$，而学界广泛推测采样对于所有$q,d\geq 3$都是可行的。迄今为止，适用于一般$q,d\geq 3$的唯一结果是由Blanca和Gheissari给出的，他们证明了随机团簇动力学在$β<β_u$时混合快速。对于$β>β_u$，由于亚稳态会出现某些相关性现象，这些现象尤其对于较小的$q$和$d$难以处理。我们的主要贡献是对Potts分布和随机团簇动力学进行超越阈值$β_u$的精细分析。我们在证明中以构造方法为主要工具，并将其与随机团簇动力学的分析相结合。因此，我们能够证明，对于所有整数$q,d\geq 3$，从全无序态初始化的随机团簇动力学在超越唯一性阈值$β_u$后均能快速混合；我们的分析一直有效到主导模态从无序相切换到有序相的阈值$β_c\in (β_u,β_u')$。我们还在有序区域$β>β_c$中获得了一种算法，该算法显著改进了$q,d$的取值范围。