Critical behaviors and critical values of branching random walks on multigraphs

We study weak and strong survival for branching random walks on multigraphs. 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 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; this improves a result of Stacey [14].