Asymptotics for the survival probability in a killed branching random walk

2008/11/03 by Nina Gantert, Gantert, Nina, Yueyun Hu +3
Mathematics · #60J80 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60J80

paper · pdf · doi:10.48550/arxiv.0811.0262

Revision

arxiv created 2010/02/16 · arxiv updated 2010/02/26

Abstract

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\- 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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