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We consider a branching random walk on $\mathbb{Z}$ started by $n$ particlesat the origin, where each particle disperses according to a mean-zero randomwalk with bounded support and reproduces with mean number of offspring$1+\theta/n$. For $t\geq 0$, we study $M_{nt}$, the rightmost position reachedby the branching random walk up to generation $[nt]$. Under certain momentassumptions on the branching law, we prove that $M_{nt}/\sqrt{n}$ convergesweakly to the rightmost support point of the local time of the limitingsuper-Brownian motion. The convergence result establishes a sharp exponentialdecay 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 speedthat is identical to that of the limiting super-Brownian motion which wasstudied 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 limitingsuper-Brownian motion, whose tail is found to decay like a Gumbel distributionwhen $\theta<0$.

On the Maximal Displacement of Near-critical Branching Random Walks