Codice QDD 49 Titolo Characterization of critical values of branching random walks on weighted graphs through infinite-type branching processes Data 2009-07-15 Autore/i Bertacchi, D. ; Zucca, F. Link Download full text Pubblicato Journal of Statistical Physics, 134 (2009), n. 1, p. 53-65 Abstract We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. Two kinds of survival can be identi¯ed: a weak survival (with positive probability there is at least one particle alive somewhere at any time) and a strong survival (with positive probability the colony survives by returning in¯nitely often to a ¯xed site). The behavior of the process depends on the value of a certain parameter which controls the birth rates; the threshold between survival and (almost sure) extinction is called critical value. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.