Asymptotics for the survival probability in a killed branching random walk

Gantert, Nina; Hu, Yueyun; Shi, Zhan

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 111-129.

Considérons une marche aléatoire branchante surcritique à temps discret. Nous nous intéressons à la probabilité qu'il existe un rayon infini du support de la marche aléatoire branchante, le long duquel elle croît plus vite qu'une fonction linéaire de pente γ - ε, où γ désigne la vitesse asymptotique de la position de la particule la plus à droite dans la marche aléatoire branchante. Sous des hypothèses générales peu restrictives, nous prouvons que, lorsque ε → 0, cette probabilité décroît comme exp{-(β+o(1)) / ε1/2}, où β est une constante strictement positive dont la valeur dépend de la loi de la marche aléatoire branchante. Dans le cas spécial où des variables aléatoires i.i.d. de Bernoulli(p) (avec 0 < p < ½) sont placées sur les arêtes d'un arbre binaire enraciné, ceci répond à une question ouverte de Robin Pemantle (Ann. Appl. Probab. 19 (2009) 1273-1291).

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, this probability decays like exp{-(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 < ½) assigned on a rooted binary tree, this answers an open question of Robin Pemantle (see Ann. Appl. Probab. 19 (2009) 1273-1291).

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