## Frogs on trees?(English)Zbl 1390.60351

Summary: 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 $$\mathrm{Pois}(\lambda)$$ 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 $$\mathcal{S}(\mathcal{T}_{d,n}):=\inf \{t:\mathcal{R}_t=\mathcal{V}_{d,n}\}$$ 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 $$d \geq 2$$ and all $$\lambda = \lambda_n>0$$ which does not diverge nor vanish too rapidly as a function of $$n$$, with high probability $$c \leq \lambda\mathcal{S}(\mathcal{T}_{d,n})/[n\log (n/\lambda)] \leq C$$ and $$\mathrm{CT}(\mathcal{T}_{d,n})\leq 3^{4\sqrt{\log |\mathcal{V}_{d,n}|}}$$.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 05C81 Random walks on graphs
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