Sharp asymptotics for the tree-completion time in cylindrical Hastings--Levitov$(0)$

Abstract

Let $\mathrm{CHL}_N$ be the cylindrical Hastings--Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $λ>0$, and let $ω_{N,λ}$ be its tree-completion time --- the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\mathbb{E}[ω_{N,λ}]\le(1+\varepsilon)(\log N)/(2λ)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[ω_{N,λ}]}{\log N}=\frac{1}{2λ} \qquad\text{for every fixed }λ>0 . \]