On the First Passage Times of Branching Random Walks in Rᵈ

We study the first passage times of discrete-time branching random walks in Rᵈ 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 ₏ (x) for general jump distributions and up to O ₏ (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.