The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908481494188032 |
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| author | Bezborodov, Viktor Gantert, Nina |
| author_facet | Bezborodov, Viktor Gantert, Nina |
| contents | We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process $\mathcal{L}$. The number of particles at time $t$ form a supercritical Galton-Watson process. We investigate the maximal distance to the origin of such branching random walks. Conditioned on survival, we show that, under some assumptions on $\mathcal{L}$, it grows in the same way as for branching Brownian motion or a broad class of one-dimensional branching random walks: the first term is linear in time and the second logarithmic. The constants in front of these terms are explicit and depend only on the mean measure of $\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem with moving barrier which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14738 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$ Bezborodov, Viktor Gantert, Nina Probability 60G50, 60J80, 60K35, 82C22 We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process $\mathcal{L}$. The number of particles at time $t$ form a supercritical Galton-Watson process. We investigate the maximal distance to the origin of such branching random walks. Conditioned on survival, we show that, under some assumptions on $\mathcal{L}$, it grows in the same way as for branching Brownian motion or a broad class of one-dimensional branching random walks: the first term is linear in time and the second logarithmic. The constants in front of these terms are explicit and depend only on the mean measure of $\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem with moving barrier which may be of independent interest. |
| title | The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$ |
| topic | Probability 60G50, 60J80, 60K35, 82C22 |
| url | https://arxiv.org/abs/2309.14738 |