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 $d\le 3$ 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 ${\mathbb{Z}}^d$, $d\le 3$. 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.

Published at http://dx.doi.org/10.1214/14-AOP947 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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