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arXiv · 2309.14738

The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$

Viktor Bezborodov·Nina Gantert
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Abstract

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.

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Viktor Bezborodov, Nina Gantert. 2025-08-07. The maximal displacement of radially symmetric branching random walk in $\mathbb{R}^d$. https://arxiv.org/abs/2309.14738

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