Asymptotics for the survival probability in a killed 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 $ε\to 0$, the probability in question decays like $\exp\{- {β+ o(1)\over ε^{1/2}}\}$, 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 $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.

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