Propagation of Branching Random Walk on Periodic Graphs

A model of a branching random walk on $\mathbb Z^d$ with a periodic set of sources of reproduction and death of particles is studied. For this model, an asymptotic description of the propagation of the population of particles with time is obtained for the first time. The intensities of the sources may be different. The branching regime is assumed to be supercritical, and the tails of the jump distribution of the walk are assumed to be “light.” The main theorem establishes the Hausdorff-metric convergence of the properly normalized random cloud of particles that exist in the branching random walk at time $t$ to the limit set, as $t$ tends to infinity. This convergence takes place for almost all elementary outcomes of the event meaning nondegeneracy of the population of particles under study. The limit set in $\mathbb R^d$, called the asymptotic shape of the population, is found in an explicit form.