Branching random walks I

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.

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

Poisson-dirichlet Branching Random Walks

We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v1, v2, . . .) where each vj is the sum of j independent Exponential(1) random variables and the different vi need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisso...