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 $n \to \infty$.
Categories: math.PR