An Improved Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

Abstract

We study the frog model on homogeneous trees, a discrete time system of simple symmetric random walks whose description is as follows. There are active and inactive particles living on the vertices. Each active particle performs a simple symmetric random walk having a geometrically distributed random lifetime with parameter (1 − p). When an active particle hits an inactive particle, the latter becomes active. We obtain an improved upper bound for the critical parameter for having indefinite survival of active particles, in the case of one-particle-per-vertex initial configuration. The main tool is to construct a class of branching processes which are dominated by the frog model and analyze their supercritical behavior. This approach allows us also to present an upper bound for the critical probability in the case of random initial configuration.