Galton--Watson 树的拟等距类
The quasi-isometry classes of Galton--Watson trees
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中文总结 AI 辅助
本文对有限支撑后代分布的 Galton-Watson 树按拟等距分类，确定射线、全树、链及茂密四类，证明类别仅依赖支撑，并给出指数尾部界及 Lean 形式化验证。
中文摘要 AI 辅助
我们对每个有限支撑后代分布的 Galton--Watson 树，按拟等距分类其大尺度几何。除平凡有限直径情形外，我们以无限直径为条件。我们发现剩余类别为：射线类、全树类（包含二叉树）、每个可能的分支半群 Λ 对应的一个链类 (C_Λ)，以及茂密类。当两棵独立树的后代分布属于同一类时，它们几乎必然存在保根拟等距；当属于不同类时，几乎必然不是拟等距的。任何两个具有有限支撑后代律的生存条件超临界实现，仍几乎必然在两个方向上都存在拟等距嵌入。我们证明实现的类别仅取决于后代分布的支撑，而非其具体分布。对于支撑在 {1,2} 上的后代分布，我们还获得了保根 D-拟等距不存在概率的显式指数尾部界。该分类还蕴含相应的随机 Cantor 边界几乎必然拟对称等价。以不灭绝为条件，这适用于强分离分形渗流，即使底层自相似迭代函数系统和保留参数不同。我们还分类了两类具有连续随机分支时间的树。我们的证明使用了新的树随机图标记的自同构匹配定理，基于失配势的收缩估计。这些二叉树上的 Markov 标记匹配结果可能具有独立意义。拟等距分类、可嵌入性和匹配定理已在 Lean 中形式化验证。
英文摘要
We classify, up to quasi-isometry, the large-scale geometry of Galton--Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the \emph{ray class}, the \emph{full tree class} (which includes the binary tree), one \emph{chain class} $(\mathrm{C}_Λ)$ for every possible branching semigroup $Λ$, and the \emph{bushy} class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely \textit{not} quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on $\{1,2\}$, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving $D$-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.
发表机构
- Fordham University（福特汉姆大学）
- Uppsala University（乌普萨拉大学）
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