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 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.

Approximating Critical Parameters of Branching Random Walks

“…On the other hand, the explicit value of λ w is not known in general. Nevertheless, in many cases it is possible to prove that λ w = 1/ lim sup n n y∈X µ (n) (x, y) (see [1] and [2]). In particular, if…”

Rumor Processes on N

“…The focus used to be on deterministic or stochastic models, modeling homogeneously mixed populations living on spaces with no structure, as in the Maki-Thompson (see [15] and [18]) and Daley-Kendall (see [5] and [17]) models. Possible variations that can be found in the recent literature include competing rumors (see [11]), more than two people meeting at a time (see [10]), moving agents (see [12]) and rumors through tree-like graphs (see [13] and [14]), complex networks (see [9]), grids (see [1]), and multigraphs (see [2]). …”

Non-intersection of transient branching random walks

“…We take X i and U i to be conditionally independent given T i for each i = 1, 2, and take the two random variables ((T 1 , o 1 ), X 1 , U 1 ) and ((T 2 , o 2 ), X 2 , U 2 ) to be independent of each other. We have by the results of [14,22] that X 1 and X 2 are both transient almost surely. Let I = X −1 1 (X 2 (V (T 2 ))).…”

“…[38,Chapter 6] for background on amenability and nonamenability. Branching random walk is particularly interesting on nonamenable graphs as it exhibits a double phase transition [11,14,22]: Suppose that µ is non-trivial. If 0 ≤ µ ≤ 1 then the process dies after finite time almost surely, if 1 < µ ≤ P −1 then the process survives forever with positive probability but does not visit any particular vertex infinitely often almost surely, while if µ > P −1 then the process has a positive probability to return to its starting point infinitely often.…”