碎片化树中的极端分离时间与进化枝计数动力学：一个冻结相变
Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition
- Technion - Israel’s Institute of Technology（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究同质碎片化树中$q$元子集的极端分离时间，证明其极限服从随机平移Gumbel分布，并揭示在阈值$\theta_*$处发生冻结相变，影响极端过程收敛及进化枝计数动力学。
AI中文摘要：
受连续时间临界β-分裂树的启发，我们研究了记录同质碎片化过程中连续块分裂及其时间的年代树，该过程限制在$n$个标签上。一个子集的分离时间是其标签首次不再位于共同块中的时间。对于每个固定整数$q\geq2$，我们通过其关联的点过程研究$q$元子集的极端分离时间。在温和条件下，当$n\to\infty$时，最后一个这样的时间在确定性中心化后具有随机平移的Gumbel极限。在阈值$\theta_*$之上，中心化中主导的$\log n$项和$\log\log n$修正项的系数变得与$q$无关，极限分布也（在确定性平移下）与$q$无关。极端过程收敛到随机平移的泊松点过程，对于$q \leq \theta_*$和$q>\theta_*$分别无装饰和有装饰。最后，在相同的中心化时间窗口内，对于$q\leq\theta_*$，大小为$q$的块（进化枝）的数量收敛到一个纯死亡过程；而对于$q>\theta_*$，则收敛到一个轨迹以正概率非单调的过程。
英文摘要：
Motivated by the continuous-time critical beta-splitting tree, we investigate chronological trees recording the successive block splits and their times in homogeneous fragmentation processes restricted to $n$ labels. The separation time of a subset is the first time its labels cease to lie in a common block. For each fixed integer $q\geq2$, we study extremal separation times of $q$-element subsets via their associated point process. Under mild conditions, as $n\to\infty$, the last such time has a randomly shifted Gumbel limit after deterministic centering. Above a threshold $θ_*$, the coefficients of both the leading $\log n$ term and the $\log\log n$ correction in the centering become independent of $q$, as does the limiting law up to deterministic translation. The extremal process converges to a randomly shifted Poisson point process, without and with decorations, for $q \le θ_*$ and $q>θ_*$ respectively. Finally, in the same centered time window, the number of size-$q$ blocks (clades) converges to a pure-death process for $q\leqθ_*$, and to a process whose trajectories are non-monotone with positive probability for $q>θ_*$.
