Capacity of the range of branching random walks in low dimensions

TL;DR

This paper investigates the growth of the capacity of the range of a branching random walk in low dimensions, showing it scales as a power of n for dimensions 3 to 5.

Contribution

It establishes the almost sure asymptotic behavior of the capacity of the range of branching random walks in dimensions 3, 4, and 5, a previously unexplored aspect.

Findings

Capacity of the range scales as n^{(d-2)/2+o(1)} for d=3,4,5.

Almost sure convergence of the capacity growth rate.

Provides new insights into the geometric properties 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 $n \to \infty$.