发表机构
Université de Lorraine, CNRS, IECL（洛林大学、法国国家科学研究中心、IECL）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对枝序图上的Aldous马尔可夫链，通过引入并行跟踪彩色子树的耦合，并利用彩色分量大小平方和的漂移性质，证明了其混合时间为$n^2$阶，解决了Aldous的猜想。
AI 中文摘要
大小为$n$的枝序图是无根二叉树，其叶子从1到$n$标记。Aldous在2000年引入了枝序图上的马尔可夫链，其一步操作包括均匀随机地移除一片叶子，并将其重新插入到均匀选择的边上。我们为此游走引入了一种耦合，该耦合并行地跟踪多个彩色子树。我们通过找到一个相关统计量来绕开彩色分量之间缺乏独立性的问题，即所有彩色分量大小的平方和，并证明该统计量具有漂移。由此我们推断出游走的混合时间为$n^2$阶，解决了Aldous的一个猜想。
英文摘要
Cladograms of size $n$ are unrooted binary trees whose leaves are labelled from 1 to $n$. Aldous introduced in 2000 a Markov chain on cladograms, a step of which consists of removing a leaf uniformly at random and reinserting it on a uniformly chosen edge. We introduce a coupling for this walk, which follows multiple colored subtrees in parallel. We go around the lack of independence of the colored components by finding a relevant statistic, namely the sum of the squares of the sizes of all colored components, which we prove has a drift. We deduce that the mixing time of the walk is of order $n^2$, solving a conjecture of Aldous.
Comments11 pages, 6 figures, comments welcome!
