THE RANGE OF TREE-INDEXED RANDOM WALK

Abstract

We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.