AI 中文总结
该研究将Beta分裂树嵌入齐次可交换碎形，通过次鞅位势测度推导各类高度渐近性质，解决了临界参数下最大高度极限的开放问题。
AI 中文摘要
受Aldous、Janson和Pittel关于临界Beta分裂模型近期研究的启发，我们通过将Beta分裂族规范地连续时间嵌入齐次可交换碎形，研究beta大于-2的完整Beta分裂族。在该表示中，标记碎形的频率由一个次鞅描述。我们用该次鞅的位势测度表示典型叶的连续高度、占据概率、离散高度及总连续时间长度。更新理论给出连续高度的一阶渐近和中心极限定理；再生组合表示给出临界值以上和临界值处离散高度的高斯极限，以及临界值以下的非高斯幂律极限。我们还利用亚纯位势理论和广义Nevanlinna函数得到位势测度和平均连续高度的留数展开式。最后，我们研究最大连续时间高度，证明其大数定律和混合Gumbel极限，在临界参数值处解决了Aldous和Janson的一个开放问题。
英文摘要
Inspired by recent work of Aldous, Janson, and Pittel on the critical beta-splitting model, we study the full beta-splitting family for beta greater than minus two through a canonical continuous-time embedding into a homogeneous exchangeable fragmentation. In this representation, the frequency of a tagged fragment is described by a subordinator. We express the continuous height of a typical leaf, its occupation probabilities, the discrete height, and the total continuous-time length in terms of the potential measure of this subordinator. Renewal theory yields first-order asymptotics and a central limit theorem for the continuous height. A regenerative-composition representation gives Gaussian limits for the discrete height above and at the critical value, and a non-Gaussian power-law limit below it. We also obtain residue expansions for the potential measure and the mean continuous height using meromorphic potential theory and generalized Nevanlinna functions. Finally, we study the maximum continuous-time height, proving a law of large numbers and a mixed Gumbel limit. At the critical parameter value, this resolves an open problem of Aldous and Janson.
Comments53 pages, 1 figure
