On the First Passage Times of Branching Random Walks in ${\mathbb{R}}^d$

Abstract

We study the first passage times of discrete-time branching random walks in ${\mathbb{R}}^d$ where $d\ge 1$. 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 $o_{\mathbb{P}}(\log x)$ for general jump distributions and up to $O_{\mathbb{P}}(\log \log x)$ 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.

41 pages, 8 figures; to appear in the Annals of Applied Probability