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 $\mathrm{CT}({\mathcal{T}}_{d,n})\le 3^{4\sqrt{\log |{\mathcal{V}}_{d,n}|}}$.

46 pages. In the latest version we allow the particle density to depend on n. A few typos corrected. To appear in Electronic Journal of Probability (2018)

References in corpus (5)

• The spread of a rumor or infection in a moving population
• From transience to recurrence with Poisson tree frogs
• Stochastic orders and the frog model
• Recurrence and Transience of Frogs with Drift on ${\mathbb{Z}}^d$
• The critical density for the frog model is the degree of the tree

Cited by in corpus (8)

• Infection spread for the frog model on trees
• Cover time for the frog model on trees
• Rapid social connectivity
• Laws of large numbers for the frog model on the complete graph
• The end time of SIS epidemics driven by random walks on edge-transitive graphs
• Multiple Random Walks on Graphs: Mixing Few to Cover Many
• A stochastic combustion model with thresholds on trees
• Critical parameter of the frog model on homogeneous trees with geometric lifetime