Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

Abstract

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 geometric 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.

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

“…We are interested in establishing, depending on the parameters, whether the process survives globally, survives locally, and if there is infinite activation or not. Local and global survival have been studied for several processes; among these it is worth mentioning the contact process and the branching random walks in continuous and discrete time (see, for instance, [3], [4], [5], [6], [12], [13], [14], and [18]).…”

Strong Local Survival of Branching Random Walks is Not Monotone

“…Most of the results of this paper still hold without this hypothesis; nevertheless, it allows us to avoid dealing with an infinite expected number of offspring. The expected number of children generated by a particle living at x is Explicit computations of M s (x, y) and M w (x) are possible in some cases (see [6] and [8]). In particular, M s (x, x) can be obtained by means of a generating function (see [28,Section 3.2]).…”

“…Example 4.1. In this example we frequently use the following argument (which is an adaptation from [6,Remark 3.2]). Consider a continuous-time BRW adapted to a connected graph X, in the sense that k xy > 0 if and only if (x, y) is an edge.…”

Approximating Critical Parameters of Branching Random Walks

“…Analogously, we denote by λ m s (X, µ) and λ m w (X, µ) (or simply by λ m s (X) and λ m w (X) or λ m s and λ m w ) the critical parameters of the BRW m on (X, µ). It is known (see, for instance, [1] and [2]) that λ s = R µ := 1/ lim sup n n µ (n) (x, y) (which is easily seen to be independent of x, y ∈ X since the graph is connected). On the other hand, the explicit value of λ w is not known in general.…”