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

TL;DR

This paper establishes the convergence in distribution of the renormalized capacity of the range of a critical branching random walk in dimensions 3 to 5, linking it to the capacity of the support of the integrated super-Brownian excursion.

Contribution

It proves the law of large numbers for the capacity of the range of critical branching random walks in specific dimensions, connecting it to super-Brownian motion.

Findings

Renormalized capacity converges in law to super-Brownian support capacity.

Convergence holds for dimensions 3, 4, and 5.

Study of intersection probabilities underpins the proof.

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 $\mathbb Z^d$.