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Asymptotics for the survival probability in a killed branching random walk. (English. French summary) Zbl 1210.60093

Summary: 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 \(\gamma - \varepsilon \), 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 \(\varepsilon \rightarrow 0\), this probability decays like \(\exp\{ - \frac{\beta +o(1)}{\varepsilon ^{1/2}} \}\), 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 \(0 < p < \frac12 \)) assigned on a rooted binary tree, this answers an open question of R. Pemantle [Ann. Appl. Probab. 19, No. 4, 1273–1291 (2009; Zbl 1176.68093)].

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1176.68093
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References:

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