Frogs on trees?

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