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.~$\mathrm{Poiss}(\lambda )$ many inactive particles are placed at each non-root 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 $\mathbf{P}(Z\ge 2)=1$ and $\mathbf{E}[Z^{4+\epsilon }]<\infty$ for some $\epsilon >0$, there is a critical value ${\lambda }_c\in (0,\infty )$ 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.

Accepted to Annals of Applied Probability

References in corpus (1)

• From transience to recurrence with Poisson tree frogs