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

Let Rn be the range of a critical branching random walk with n particles on Zd, 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 d∈{3,4,5}, we prove that n−d−24cap(d)(Rn), the renormalized capacity of Rn, 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 Zd.