Capacity of the Range of Branching Random Walks in Low Dimensions

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