Eyal Neumann

On the Maximal Displacement of Near-critical Branching Random Walks

We consider a branching random walk on the one-dimensional integer lattice, started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring 1+\theta/n. For t >0, we study M_{nt}, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that M_{nt}/n^{1/2} converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of M_{nt}. We also confirm that when \theta>0, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky in [1]. The rightmost position over all generations, M:=sup_t M_{nt}, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when \theta<0. This is a joint work with Xinghua Zheng. [1] R. G. Pinsky. On the large time growth rate of the support of supercritical super-Brownian motion. Ann. Probab., 23(4):1748-1754, 1995.