Shape theorem for excited random walks on the integer lattice

Abstract

We study the limit shape of the range of the random walk excited to the center on $\mathbb Z^d$, with $d\geq2$. A cookie is placed at each nonzero vertex. On its first visit to such a vertex, the walker consumes the cookie and takes its next nearest-neighbor step with a bias toward the origin. On a visit to the origin, or on later visits to a nonzero vertex, no cookie is available and the walker moves to a neighbor uniformly at random. We prove that the set of vertices visited up to time $n$, rescaled by $n^{-1/(d+1)}$, converges in probability in Hausdorff distance to an explicit $\ell^1$ ball. This answers a question raised by Kozma (2007).