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

TL;DR

This paper analyzes the asymptotic behavior of the largest and smallest fragments in a $k$-regular self-similar fragmentation process, establishing precise bounds and distributional limits for fragmentation times.

Contribution

It introduces explicit deterministic functions approximating the sizes of extremal fragments and links the process to branching random walks to derive asymptotic laws.

Findings

Largest and smallest fragment sizes are tightly bounded by deterministic functions.

Fragmentation times for small fragments converge to a Gumbel distribution after rescaling.

The process is effectively related to branching random walks and point processes.

Abstract

We study the asymptotics of the $k$-regular self-similar fragmentation process. For $\alpha > 0$ and an integer $k \geq 2$, this is the Markov process $(I_t)_{t \geq 0}$ 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 $(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq 1$ and $|M_t - h(t)| \leq 1$ 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 $n \to \infty$.