The range of tree-indexed random walk in low dimensions

Abstract

We study the range $R_{n}$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^{d}$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_{n}$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in $\mathbb{Z}^{d}$, $d\leq3$. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.