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

TL;DR

This paper analyzes the maximal displacement in radially symmetric branching random walks in Euclidean space, showing it grows linearly with time plus a logarithmic correction, with explicit constants depending on offspring distribution and dimension.

Contribution

It establishes the asymptotic growth of maximal displacement for radially symmetric branching random walks, extending known results to higher dimensions with explicit constants.

Findings

Maximal displacement grows linearly with time.

A logarithmic correction term is present.

Explicit constants depend on offspring distribution and dimension.

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.