Work

On the first passage times of branching random walks in Rd

Year: 2026

Type: article

Abstract: 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 at a distance x from the origin, conditioned upon survival. We explicitly provide the linear dominating term and the logarithmic correction term as functions of x. The asymptotics are precise up to an order of oP(logx) for general jump distributions and up to OP(loglogx) 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.

Source: The Annals of Applied Probability

DOI: 10.1214/26-aap2296

Authors: Jose Blanchet, Wei Cai, Shaswat Mohanty, Zhenyuan Zhang

Institution: Stanford University

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Robust Wasserstein Profile Inference, DMS-EPSRC: Fast Martingales, Large Deviations, and Randomized Gradients for Heavy-tailed Distributions, Collaborative Research: CIF: Medium: Statistical and Algorithmic Foundations of Distributionally Robust Policy Learning, Collaborative Research: AMPS: Rare Events in Power Systems: Novel Mathematics, Statistics and Algorithms.