Linear and superlinear spread for continuous-time frog model

Abstract

Consider a stochastic growth model on $\mathbb{Z} ^d$. Start with some active particle at the origin and sleeping particles elsewhere. The initial number of particles at $x \in \mathbb{Z} ^d$ is $\eta(x)$, where $\eta (x)$ are independent random variables distributed according to $\mu$. Active particles perform a simple continuous-time 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 proofs rely on comparison to various percolation models.