Blanchet, J., Cai, W., Mohanty, S., & Zhang, Z. (2024). On the First Passage Times of Branching Random Walks in $\mathbb R^d$. ArXiv. /abs/2404.09064

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Abstract

We study the first passage times of discrete-time branching random walks in where . 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 from the origin, conditioned upon survival. We provide explicitly the linear dominating term and the logarithmic correction term as a function of . The asymptotics are precise up to an order of for general jump distributions and up to 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.

Authors
Jose Blanchet, Wei Cai, Shaswat Mohanty, Zhenyuan Zhang
Publication date
2024/4/13
Journal
arXiv preprint arXiv:2404.09064