Asymptotics for the survival probability in a killed branching random walk

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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 {- β+ o(1) 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<1 2) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.