Tightness Analysis of First Passage Times of $d$-Dimensional Branching Random Walk
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866918435963797504 |
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| author | Blanchet, Jose Zhang, Zhenyuan |
| author_facet | Blanchet, Jose Zhang, Zhenyuan |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tightness Analysis of First Passage Times of $d$-Dimensional Branching Random Walk Blanchet, Jose Zhang, Zhenyuan Probability 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. |
| title | Tightness Analysis of First Passage Times of $d$-Dimensional Branching Random Walk |
| topic | Probability |
| url | https://arxiv.org/abs/2410.02635 |