2019/10/06 by Michelen, Marcus, Rosenberg, Josh
#60K35 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1910.02367
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.~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 P(Z ≥ 2) = 1 and E[Z4 + ε] < ∞ for some ε> 0, there is a critical value λc∈(0,∞) 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.