Frogs on trees?

Abstract

We study a system of simple random walks on $\mathcal{T}_{d,n} = \mathcal{V}_{d,n}, \mathcal{E}_{d,n})$, 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) $ t \in \mathbb{N} \cup \{\infty \} $ steps of simple random walk on the tree. 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. Let $\mathcal{S}(\mathcal{T}_{d,n}) := \inf \{t:\mathcal{R}_t = \mathcal{V}_{d,n} \} $. Let the cover time $\mathrm{CT}(\mathcal{T}_{d,n})$ be 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 $ which does not diverge nor vanish too rapidly, with high probability $c \le \lambda \mathcal{S}(\mathcal{T}_{d,n}) /n \log (n/\lambda) \le C$ and $\mathrm{CT}(\mathcal{T}_{d,n}) \le 3^{4 \sqrt{ \log |\mathcal{V}_{d,n}| }}$.