Convergence in law for the capacity of the range of a critical branching random walk

Abstract

Let ${\mathit{R}}_{\mathit{n}}$ be the range of a critical branching random walk with n particles on ${\mathbb{Z}}^{\mathit{d}}$, which is the set of sites visited by a random walk indexed by a critical Galton–Watson tree conditioned on having exactly n vertices. For $\mathit{d}\in \{3,4,5\}$, we prove that ${\mathit{n}}^{-\frac{\mathit{d}-2}{4}}{\mathtt{cap}}^{(\mathit{d})}({\mathit{R}}_{\mathit{n}})$, the renormalized capacity of ${\mathit{R}}_{\mathit{n}}$, converges in law to the capacity of the support of the integrated super-Brownian excursion. The proof relies on a study of the intersection probabilities between the critical branching random walk and an independent simple random walk on ${\mathbb{Z}}^{\mathit{d}}$.