Extremal separation times and clade-count dynamics in fragmentation trees: a freezing transition

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 $q\ge 2$, we study extremal separation times of $q$-element subsets via their associated point process. Under mild conditions, as $n\to \mathrm{\infty }$, the last such time has a randomly shifted Gumbel limit after deterministic centering. Above a threshold ${\theta }_{\ast }$, 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 $q\le {\theta }_{\ast }$ and $q>{\theta }_{\ast }$ respectively. Finally, in the same centered time window, the number of size-$q$ blocks (clades) converges to a pure-death process for $q\le {\theta }_{\ast }$, and to a process whose trajectories are non-monotone with positive probability for $q>{\theta }_{\ast }$.