On the First Passage Times of Branching Random Walks in ℝ^d
arxiv(2024)
摘要
We study the first passage times of discrete-time branching random walks in
ℝ^d where d≥ 1. Here, the genealogy of the particles follows a
supercritical Galton-Watson process. We provide asymptotics of the first
passage times to a ball of radius one with a distance x from the origin,
conditioned upon survival. We provide explicitly the linear dominating term and
the logarithmic correction term as a function of x. The asymptotics are
precise up to an order of o_ℙ(log x) for general jump
distributions and up to O_ℙ(loglog x) for spherically symmetric
jumps. A crucial ingredient of both results is the tightness of first passage
times. We also discuss an extension of the first passage time analysis to a
modified branching random walk model that has been proven to successfully
capture shortest path statistics in polymer networks.
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