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.~$\mathrm{Poiss}(λ)$ 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 $\mathbf{P}(Z \geq 2) = 1$ and $\mathbf{E}[Z^{4 + ε}] < \infty$ for some $ε> 0$, there is a critical value $λ_c\in(0,\infty)$ 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.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof