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

Research output: Contribution to journalArticlepeer-review

Abstract

We study the asymptotics of the k-regular self-similar fragmentation process. For α > 0 and an integer k ≥ 2, this is the Markov process (It )t≥0 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 k −mt and k −mt be the respective sizes of the largest and smallest fragments in It. By relating (It )t≥0 to a branching random walk, we find that there exist explicit deterministic functions g(t) and h(t) such that |mt − g(t)| ≤ 1 and |Mt −h(t)| ≤ 1 for all sufficiently large t. Furthermore, for each n, we study the final time at which fragments of size k −n 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→∞.

Original languageEnglish
Pages (from-to)1173-1203
Number of pages31
JournalAnnals of Probability
Volume50
Issue number3
DOIs
StatePublished - May 2022

Keywords

  • Branching random walk
  • Fragmentation
  • Point process

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