Sampling graph colorings via local Markov chains is a central problem in approximate counting and Markov chain Monte Carlo (MCMC). We address the problem of sampling a random $k$-coloring of a graph with maximum degree $Δ$. The simplest algorithmic approach is to establish rapid mixing of the single-site update chain known as the Metropolis Glauber dynamics, which at each step chooses a random vertex $v$ and proposes a random color $c$, recoloring $v$ to $c$ if the resulting coloring remains proper. It is a long-standing open problem to prove that the Glauber dynamics has polynomial mixing time on all graphs whenever $k\geqΔ+2$. We prove that for every $δ>0$ and all $Δ\geq Δ_0(δ)$, if $k\ge (1+δ)Δ$ then the Glauber dynamics has optimal mixing time of $O_δ(|V| \log |V|)$ on any graph of girth $\geq 11$ and maximum degree $Δ$. Our approach builds on a non-Markovian coupling introduced by Hayes and Vigoda (2003) for the large-degree regime $Δ=Ω(\log n)$, in which updates at time $t$ may depend on and modify proposed updates at future times. A complete analysis of this framework requires resolving substantial technical obstacles that remain in the original argument, and extending it to the constant-degree regime introduces further difficulties, since non-Markovian updates may fail with constant probability. We overcome these obstacles by developing and analyzing a refined local non-Markovian coupling, and by establishing new local-uniformity results for the Metropolis dynamics, extending prior results for the heat-bath chain due to Hayes (2013). Together, these ingredients provide a complete analysis of the non-Markovian coupling framework in the large-degree regime, while simultaneously strengthening it substantially to obtain optimal mixing all the way down to the constant-degree setting.

翻译：通过局部马尔可夫链对图着色进行采样是近似计数和马尔可夫链蒙特卡洛方法中的一个核心问题。我们研究从最大度为$Δ$的图中随机采样一个$k$-着色的问题。最简单的算法方法是证明单点更新链（即Metropolis Glauber动力学）的快速混合性，该链在每一步随机选择一个顶点$v$并提议一个随机颜色$c$，若得到的着色仍为合法着色，则将$v$重染为$c$。一个长期未解决的公开问题是：证明当$k\geqΔ+2$时，Glauber动力学在所有图上都具有多项式混合时间。我们证明：对任意$δ>0$和所有$Δ\geq Δ_0(δ)$，若$k\ge (1+δ)Δ$，则对于任意围长$\geq 11$且最大度为$Δ$的图，Glauber动力学具有最优混合时间$O_δ(|V| \log |V|)$。我们的方法建立在Hayes与Vigoda（2003）针对大度数区域$Δ=Ω(\log n)$提出的非马尔可夫耦合之上，该耦合中时间$t$的更新可能依赖于并修改未来时间的提议更新。对这一框架的完整分析需要解决原始论证中遗留的重大技术障碍，而将其推广到常数度数区域会引入进一步困难，因为非马尔可夫更新可能以常数概率失败。我们通过开发并分析一种精细的局部非马尔可夫耦合，以及建立Metropolis动力学的新局部均匀性结果（扩展了Hayes（2013）针对热浴链的前期结果）来克服这些障碍。这些要素共同实现了对大度数区域非马尔可夫耦合框架的完整分析，同时显著增强了该框架，从而在下至常数度数设置中也获得最优混合时间。