Minimal displacement of branching random walk

Let {Mathematical expression} denote a branching random walk in {Mathematical expression} with mean particle production m, m>1, and with incremental spatial distribution G, with G({0}) =p and G({1})=1-p. If mp=1, then the minimal displacement of {Mathematical expression} behaves asymptotically like log log n/log 2. If the condition G({1})=1-p is replaced by G((0, ∞))=1-p, we obtain a similar result.