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

Abstract

Let $R_n$ be the range of a critical branching random walk with $n$ particles on ${\mathbb{Z}}^d$, 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 $d\in \{3,4,5\}$, we prove that $n^{-\frac{d-2}{4}}{\mathtt{cap}}^{(d)}(R_n)$, the renormalized capacity of $R_n$, 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 ${\mathbb{Z}}^d$.

arXiv commentThere was a mistake in the proof of Theorem 4.1, now solved by an improvement of Corollary 4.14