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 $\lambda >0$, and let ${\omega }_{N,\lambda }$ 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 $\mathbb{E}[{\omega }_{N,\lambda }]\le (1+\epsilon )(\log N)/(2\lambda )$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[ω_{N,λ}]}{\log N}=\frac{1}{2λ} \qquad\text{for every fixed }λ>0 . \]

10 pages, 1 figure