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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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