Moments and distribution of the local times of a transient random walk on ${\mathbb{Z}}^d$

Abstract

Consider an arbitrary transient random walk on ${\mathbb{Z}}^d$ with $d\in \mathbb{N}$. Pick $\hat{I}\pm \in [0,\infty )$ and let $L_n(\hat{I}\pm )$ be the spatial sum of the $\hat{I}\pm$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $\hat{I}\pm$, $L_n(\hat{I}\pm )$ is the number of the $\hat{I}\pm$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(\hat{I}\pm )$ as $n\to \infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý \cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.

9 pages