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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.02635 |
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Table of Contents:
- Given a discrete-time non-lattice supercritical branching random walk in $\mathbb{R}^d$, we investigate its first passage time to a shifted unit ball of a distance $x$ from the origin, conditioned upon survival. We provide precise asymptotics up to $O(1)$ (tightness) for the first passage time as a function of $x$ as $x\to\infty$, thus resolving a conjecture in Blanchet--Cai--Mohanty--Zhang (2024). Our proof builds on the previous analysis of Blanchet--Cai--Mohanty--Zhang (2024) and employs a careful multi-scale analysis on the genealogy of particles within a distance of $\asymp \log x$ near extrema of a one-dimensional branching random walk, where the cluster structure plays a crucial role.