Frogs on trees?

Abstract

We study a system of simple random walks on $\mathcal{T} _{d,n}=({\cal V}_{d,n},{\cal E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda $) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o} $. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o} $. Active particles perform independent simple random walk on the tree of length $ t \in{\mathbb N} \cup \{\infty \} $, referred to as the particles’ lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R} _t$ be the set of vertices which are visited by the process (with lifetime $t$). The susceptibility ${\mathcal S}({\mathcal T}_{d,n}):=\inf \{t:\mathcal{R} _t={\cal V}_{d,n} \} $ is the minimal lifetime required for the process to visit all sites. The cover time $\mathrm{CT} ({\mathcal T}_{d,n})$ is the first time by which every vertex was visited at least once, when we take $t=\infty $. We show that there exist absolute constants $c,C>0$ such that for all $d \ge 2$ and all $\lambda = {\lambda }_n >0$ which does not diverge nor vanish too rapidly as a function of $n$, with high probability $c \le \lambda{\mathcal S} ({\mathcal T}_{d,n}) /[n\log (n / {\lambda } )] \le C$ and $\mathrm{CT} ({\mathcal T}_{d,n})\le 3^{4\sqrt{ \log |{\cal V}_{d,n}| } }$.