On the maximal displacement of near-critical branching random walks

We consider a branching random walk on $$\mathbb {Z}$$ 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 $$1+\theta /n$$ 1 + θ / n . For $$t\ge 0$$ t ≥ 0 , we study $$M_{nt}$$ 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 $$M_{nt}/\sqrt{n}$$ M nt / 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 $$M_{nt}$$ M nt . We also confirm that when $$\theta >0$$ 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, $$M:=\sup _t M_{nt}$$ 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 $$\theta <0$$ θ < 0 .