Extinction window of mean field branching annihilating random walk

Abstract

We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.Comment: Published at http://dx.doi.org/10.1214/14-AAP1069 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Phase transition in a double branching annihilating random walk

“…This lack of monotonicity practically prevents us to compare these processes via domination arguments. Hence the long-time behavior such as the survival or extinction seem to be difficult to treat in general (see [5,15,6,16,3,17,4,13] and further references therein).…”