May 2022 Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation
Piotr Dyszewski, Nina Gantert, Samuel G. G. Johnston, Joscha Prochno, Dominik Schmid
Author Affiliations +
Ann. Probab. 50(3): 1173-1203 (May 2022). DOI: 10.1214/21-AOP1556


We study the asymptotics of the k-regular self-similar fragmentation process. For α>0 and an integer k2, this is the Markov process (It)t0 in which each It is a union of open subsets of [0,1), and independently each subinterval of It of size u breaks into k equally sized pieces at rate uα. Let kmt and kMt be the respective sizes of the largest and smallest fragments in It. By relating (It)t0 to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that |mtg(t)|1 and |Mth(t)|1 for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size kn 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 n.

Funding Statement

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.


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Piotr Dyszewski. Nina Gantert. Samuel G. G. Johnston. Joscha Prochno. Dominik Schmid. "Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation." Ann. Probab. 50 (3) 1173 - 1203, May 2022.


Received: 1 February 2021; Revised: 1 October 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

MathSciNet: MR4413214
zbMATH: 1498.60315
Digital Object Identifier: 10.1214/21-AOP1556

Primary: 60G55 , 60J27 , 60J80

Keywords: Branching random walk , fragmentation , point process

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • May 2022
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