Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition
Abstract
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≥2, we study extremal separation times of q-element subsets via their associated point process. Under mild conditions, as n→∞, 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 loglog 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 ≤ 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≤θ*, and to a process whose trajectories are non-monotone with positive probability for q>θ*.
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BibTeX
@misc{ma2026extremal,
title = {{Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition}},
author = {Heng Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14325},
eprint = {2609.14325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14325},
}
RIS
TY - UNPB
TI - Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition
AU - Heng Ma
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.14325
UR - https://arxiv.org/abs/2609.14325
AB - 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≥2, we study extremal separation times of q-element subsets via their associated point process. Under mild conditions, as n→∞, 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 loglog 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 ≤ 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≤θ_*, and to a process whose trajectories are non-monotone with positive probability for q>θ_*.
ER -