扩张图上的重复平均 I：随机 $d$-正则图
Repeated averaging on expanders I: random $d$-regular graph
- Jiangsu Normal University（江苏师范大学）
- Caltech（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
针对一般图上悬而未决的$L^1$截止问题，本文提出基于大小偏置抽样的判据，证明随机$d$-正则图（$d\ge 3$）上存在高斯轮廓的$L^1$截止，且截止时间晚于随机游走和完全图下界，揭示熵间隙。
中文摘要 AI 辅助
图上的重复平均是局部随机交换的自然原型，用于模拟共识形成、信息传播和财富再分配。其收敛性长期以来备受关注，从80年代Bourgain关于完全图$L^1$截止的问题（近年来由Chatterjee、Diaconis、Sly和Zhang解答）到Aldous和Lanoue的系统研究。针对一般图上仍然悬而未决且被广泛询问的$L^1$截止问题，我们建立了一个基于大小偏置抽样的判据，并为稀疏扩张图发展了一个通用框架。具体而言，我们证明了在随机$d$-正则图上，对于任意$d\ge 3$且均匀覆盖起始顶点，$L^1$截止具有高斯轮廓。值得注意的是，尽管期望质量轮廓与相应的随机游走分布完全一致，但截止发生的时间严格晚于随机游走截止时间和完全图上达到的通用下界，揭示了意想不到的熵间隙。该框架是稳健的，涉及聚合物表示、随机几何、万有覆盖提升和自适应揭示等思想。我们的后续工作将将其扩展到Ramanujan图和有限度配置模型。
英文摘要
Repeated averaging on graphs is a natural prototype of local stochastic exchange, modeling consensus formation, information spreading, and wealth redistribution. Its convergence has attracted longstanding interest, from Bourgain's complete-graph $L^1$ cutoff question in the 80s (answered in recent years by Chatterjee, Diaconis, Sly, and Zhang) and the systematic study of Aldous and Lanoue. Addressing the still outstanding and widely asked $L^1$-cutoff question on general graphs, we establish a criterion based on size-biased sampling, and develop a general framework for sparse expanders. Specifically, we prove $L^1$ cutoff with a Gaussian profile on random $d$-regular graphs, for any $d\ge 3$ and uniformly over starting vertices. Remarkably, although the expected mass profile agrees exactly with the corresponding random-walk distribution, cutoff occurs strictly later than both the random walk cutoff time and the universal lower bound attained on complete graphs, revealing an unexpected entropy gap. The framework is robust, involving ideas of polymer representations, random geometry, universal-cover lifting, and adaptive revealing. Our companion work will extend it to Ramanujan graphs and bounded-degree configuration models.
