On the First Passage Times of Branching Random Walks in $\mathbb R^d$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908745038036992 |
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| author | Blanchet, Jose Cai, Wei Mohanty, Shaswat Zhang, Zhenyuan |
| author_facet | Blanchet, Jose Cai, Wei Mohanty, Shaswat Zhang, Zhenyuan |
| contents | We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. 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 $o_{\mathbb P}(\log x)$ for general jump distributions and up to $O_{\mathbb P}(\log\log x)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_09064 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the First Passage Times of Branching Random Walks in $\mathbb R^d$ Blanchet, Jose Cai, Wei Mohanty, Shaswat Zhang, Zhenyuan Probability 60G70, 60J80, 60J85, 60G50 We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. 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 $o_{\mathbb P}(\log x)$ for general jump distributions and up to $O_{\mathbb P}(\log\log x)$ 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. |
| title | On the First Passage Times of Branching Random Walks in $\mathbb R^d$ |
| topic | Probability 60G70, 60J80, 60J85, 60G50 |
| url | https://arxiv.org/abs/2404.09064 |