The largest and smallest fragment in a k-regular self-similar fragmentation

We study the asymptotics of the $k$-regular self-similar fragmentation process. For $\alpha \geq 0$ 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 $(I_t)_{\geq 0}$ 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 $t$. The talk is based on a joint work with Nina Gantert, Samuel G. G. Johnston, Joscha Prochno and Dominik Schmid.