Linear and superlinear spread for continuous-time frog model

Viktor Bezborodov, Tyll Krueger

Published 2020-08-24Version 1

We consider a stochastic growth model on $\mathbb{Z} ^{\rm{d}}$. Start with some active particle at the origin and $\eta(x)$ sleeping particles at $x \in \mathbb{Z} ^{\rm{d}}$, where $\eta (x)$ are independent random variables distributed according to $\mu$. Active particles perform a continuous-time simple random walk while sleeping particles stay put until the first arrival of an active particle to their location. Upon the arrival all sleeping particles at the site activate at once and start moving according to their own simple random walks. The aim of this paper is to give conditions on $\mu$ under which the spread of the process is linear or faster than linear. The main technique is comparison with other stochastic growth models.