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 $\alpha >0$ and an integer $k\ge 2$, this is the Markov process $(I_t{)}_{t\ge 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^{\alpha }$. Let $k^{-m_t}$ and $k^{-M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t{)}_{t\ge 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t-g(t)|\le 1$ and $|M_t-h(t)|\le 1$ 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 $n\to \infty$.

29 pages, 1 figure