Scaling limits of tree-valued branching random walks

Branching Random Walks

We study a generalized branching random walk where particles breed at a rate which depends on the number of neighbouring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a non trivial invariant measure for this case.

Maxima of Branching Random Walks

In recent years several authors have obtained limit theorems for Ln, the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has (p(O)=.fexp(Ox)dF(x)0. In this paper we investigate what happens when there is a slowly varying function K so that 1-F(x)~x-qK(x) as x...

Branching Random Walks on Graphs

We study a new distributed randomized information propagation mechanism in networks that we call a branching random walk (BRW). BRW is a generalization of the well-studied “standard” random walk which is a fundamental primitive useful in a wide variety of network applications ranging from token management and load balancing to search, routing, information propagation and gossip. BRW is paramete...

Minima in Branching Random Walks

Given a branching random walk, let Mn be the minimum position of any member of the n’th generation. We calculate EMn to within O(1) and prove exponential tail bounds for P {|Mn − EMn| > x}, under quite general conditions on the branching random walk. In particular, together with work of [8], our results fully characterize the possible behavior of EMn when the branching random walk has bounded b...

Branching Processes and Random Walks