Capacity of the range of branching random walks in low dimensions

Abstract

Consider a branching random walk $(V_u{)}_{u\in {\mathcal{T}}^{IGW}}$ in ${\mathbb{Z}}^d$ with the genealogy tree ${\mathcal{T}}^{IGW}$ formed by a sequence of i.i.d. critical Galton-Watson trees. Let $R_n$ be the set of points in ${\mathbb{Z}}^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in ${\mathcal{T}}^{IGW}$. Our main result states that for $d\in \{3,4,5\}$, the capacity of $R_n$ is almost surely equal to $n^{\frac{d-2}{2}+o(1)}$ as $n\to \infty$.

arXiv commentThe final version of the paper is published in Proc. Steklov Inst. Math. 316 (2022) http://link.springer.com/journal/11501/316/1/page/1