Asymptotics for the survival probability in a supercritical branching random walk

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope γ − ε, where γ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when ε → 0, the probability in question decays like exp{− ε1/2 }, where β is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli(p) random variables (with 0 < p < 12) assigned on a rooted binary tree, this answers an open question of Robin Pemantle [10].