On the Maximal Displacement of Near-critical Branching Random Walks

We consider a branching random walk on $\mathbb{Z}$ 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 $1+\theta/n$. For $t\geq 0$, 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 $M_{nt}/\sqrt{n}$ 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 $M_{nt}$. 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 [28]. 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 $\theta<0$.