We analyse uniformly random proper $k$-colourings of sparse graphs with maximum degree $\Delta$ in the regime $\Delta < k\ln k $. This regime corresponds to the lower side of the shattering threshold for random graph colouring, a paradigmatic example of the shattering threshold for random Constraint Satisfaction Problems. We prove a variety of results about the solution space geometry of colourings of fixed graphs, generalising work of Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the performance of stochastic local search algorithms in this regime. Our central proof relies only on elementary techniques, namely the first-moment method and a quantitative induction, yet it strengthens list-colouring results due to Vu, and more recently Davies, Kang, P., and Sereni, and generalises state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It further yields an approximately tight lower bound on the number of colourings, also known as the partition function of the Potts model, with implications for efficient approximate counting.

翻译：我们分析了最大度 $\Delta$ 满足 $\Delta < k\ln k$ 的稀疏图上的一致随机正常 $k$-着色。这一区域对应于随机图着色中破碎阈值（随机约束满足问题的破碎阈值的典型例子）的较低侧。我们证明了关于固定图着色解空间几何性质的多种结果，推广了Achlioptas、Coja-Oghlan和Molloy在随机图上的工作，并论证了该区域内随机局部搜索算法的性能。我们的核心证明仅依赖于初等方法，即一阶矩方法和定量归纳法，但它强化了Vu以及近期Davies、Kang、P.和Sereni的列表着色结果，并推广了稀疏图背景下拉姆齐理论的最新界。此外，它还为着色数量（也称为Potts模型的配分函数）提供了近似紧的下界，这对高效近似计数具有启示意义。