Asymptotics for the survival probability in a killed branching random walk
Résumé
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 gamma - epsilon, where gamma 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 epsilon -> 0, this probability decays like exp{-beta+o(1)/epsilon(1/2)}, where beta 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 (see Ann. Appl. Probab. 19 (2009) 1273-1291).