Bounds on the critical parameter for the frog model on trees

Abstract

We consider the frog model with a lifetime on infinite trees. In this model, frogs (particles), when awake, perform a symmetric random walk on the tree, waking up all dormant frogs at the sites visited until they die. We consider variations of the model by changing the structure of the tree (oriented or not, random or not) and the survival distribution. In these models, the survival probability of a frog is controlled by a parameter \( p \in [0,1] \), and there is a critical value \( p_c \) such that if \( p < p_c \), then only finitely many frogs are awakened with probability 1, while if \( p > p_c \), infinitely many frogs are awakened with positive probability. The thesis is dedicated to obtaining lower and/or upper bounds for this critical parameter as a function of structural constants of the considered models.