Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation

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→∞.