We study the asymptotics of the k-regular self-similar fragmentation process. For and an integer , this is the Markov process in which each is a union of open subsets of , and independently each subinterval of of size u breaks into k equally sized pieces at rate . Let and be the respective sizes of the largest and smallest fragments in . By relating to a branching random walk, we find that there exist explicit deterministic functions and such that and for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as .
SJ and JP are supported by the Austrian Science Fund (FWF) Project P32405 Asymptotic geometric analysis and applications of which JP is principal investigator.
DS thanks the Studienstiftung des deutschen Volkes and the TopMath program for financial support.
The research of PD was supported by the Alexander von Humboldt Foundation.
We thank Günter Last for answering questions about point processes and two anonymous referees for a careful reading and helpful suggestions.
"Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation." Ann. Probab. 50 (3) 1173 - 1203, May 2022. https://doi.org/10.1214/21-AOP1556