One-arm domination time in Cylindrical Hastings-Levitov$(0)$

Abstract

The cylindrical Hastings-Levitov$(0)$ admits a single infinite connected tree (arm). For a cylinder of width $N$ and particles of size $\lambda$, {we consider the first time $\upsilon_{N, \lambda}$ after which only the unique infinite tree receives particles}. We prove that $\frac{cN^2}{\lambda^3} \le \mathbb{E}[\upsilon_{N, \lambda}]\le\frac{CN^2}{\lambda^3}$, and establish an exponential tail for $\upsilon_{N, \lambda}$. Moreover, we obtain an asymptotic bound to the expected total number of trees, and the last time a new tree emerges.