The frog model on Galton–Watson trees

Abstract

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d. $\mathrm{Poiss}(\mathit{\lambda })$ many inactive particles are placed at each nonroot vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton–Watson trees with offspring distributions Z satisfying $\mathbf{P}(\mathit{Z}\ge 2)=1$ and $\mathbf{E}[{\mathit{Z}}^{4\mathbf{+}\mathit{\epsilon }}]<\infty$ for some $\mathit{\epsilon }>0$, there is a critical value ${\mathit{\lambda }}_{\mathit{c}}\in (0,\infty )$ separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman–Johnson–Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of Z, answering another question of Hoffman–Johnson–Junge and showing that the frog model and contact process behave differently on Galton–Watson trees.