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