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 $\frac{c{N}^2}{{\lambda }^3}\le \mathbb{E}[{\upsilon }_{N,\lambda }]\le \frac{C{N}^2}{{\lambda }^3}$, 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.

30 pages