Critical behaviors and critical values of branching random walks on multigraphs

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometrical parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs) we prove that, at the weak critical value, the process dies out globally almost surely. Moreover for the same class we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.