On the maximal displacement of near-critical branching random walks

Abstract

We consider a branching random walk on Z 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 + θ/ n. For t≥ 0 , we study Mnt, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that Mnt/n 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 Mnt. We also confirm that when θ> 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 (Ann Probab 23(4):1748–1754, 1995). The rightmost position over all generations, M: = sup tMnt, 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 θ< 0.