AI 中文总结
该研究将正则树的低温Potts模型Glauber动力学近线性混合结果扩展到泊松树，提升了稀疏随机图上Potts模型近似采样算法的效率。
AI 中文摘要
图$G$上的$q$态铁磁Potts模型是$G$的所有$q$着色上的概率分布，倾向于产生大量同色边。对Potts模型进行近似采样是稀疏图上自旋系统研究的核心问题，尤其在低温 regime（ regime 可译为“ regime”，保留原词），该模型强烈倾向于有序构型，常形成瓶颈导致马尔可夫链采样效率低下或难以分析。我们聚焦于稀疏随机图$G(n,d/n)$，其局部邻域呈树状，但相关底层图是泊松Galton-Watson树，这促使我们研究具有同色边界条件的此类树上低温Potts模型的Glauber动力学。泊松设定引入了正则情形不存在的困难：度数波动、可能出现长诱导路径、分支可能在到达边界前终止，导致叶节点处同色边界的影响远非均匀。我们的主要结果表明，在具有同色边界条件的泊松树上，Glauber动力学具有近线性混合特性，这将Martinelli、Sinclair和Weitz（SODA 2004）以及Blanca、Chen、Stefankovič和Vigoda（RANDOM 2021）的相应正则树结果扩展到了稀疏随机图产生的非正则树。我们的证明引入了围绕包含大型正则子树的区域构建的树的自适应块分解，并将其与相关性衰减估计和函数不等式论证相结合。我们还获得了$G(n,d/n)$上所有温度下Potts模型的近线性时间近似采样算法，该算法比Galanis、Goldberg和Smolarova（ICALP 2025）的最佳先前算法更快，其主要新要素是基于泊松树结果对低温 regime 的精细分析。
英文摘要
The $q$-state ferromagnetic Potts model on a graph $G$ is a probability distribution on all $q$-colourings of $G$ that favours many monochromatic edges. Approximate sampling from the Potts model is a central problem in the study of spin systems on sparse graphs, especially in the low-temperature regime, where the model strongly favours ordered configurations, often creating bottlenecks that make Markov-chain sampling inefficient or difficult to analyse. We focus on the sparse random graph $G(n,d/n)$. The local neighbourhoods of $G(n,d/n)$ are tree-like, but the relevant underlying graph is a Poisson Galton-Watson tree. This motivates the study of Glauber dynamics for the low-temperature Potts model on such trees with monochromatic boundary conditions. The Poisson setting introduces difficulties absent from the regular case: degrees fluctuate, long induced paths may appear, and branches can terminate before reaching the boundary. As a result, the effect of the monochromatic boundary at the leaves is much less uniform. Our main result shows near-linear mixing for the Glauber dynamics on Poisson trees with monochromatic boundary conditions. This extends the corresponding regular-tree results of Martinelli, Sinclair, and Weitz (SODA 2004) and of Blanca, Chen, Stefankovič, and Vigoda (RANDOM 2021) to the irregular trees arising from sparse random graphs. Our proof introduces an adaptive block decomposition of the tree, built around regions containing large regular subtrees, and combines it with correlation-decay estimates and functional-inequality arguments. We also obtain a near-linear-time approximate sampling algorithm for the Potts model on $G(n,d/n)$ at all temperatures, speeding up the best previous algorithm of Galanis, Goldberg, and Smolarova (ICALP 2025). The main new ingredient is a refined analysis of the low-temperature regime, building on the Poisson tree result.
CommentsAbstract shortened to meet arXiv submission requirements
