PreprintarXiv licencearXiv
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 ℤd, with d≥2. 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 ℓ1 ball. This answers a question raised by Kozma (2007).
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BibTeX
@misc{nguyen2026shape,
title = {{Shape theorem for excited random walks on the integer lattice}},
author = {Tuan-Minh Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07724},
eprint = {2610.07724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07724},
}
RIS
TY - UNPB
TI - Shape theorem for excited random walks on the integer lattice
AU - Tuan-Minh Nguyen
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07724
UR - https://arxiv.org/abs/2610.07724
AB - We study the limit shape of the range of the random walk excited to the center on ℤ^d, with d≥2. 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 ℓ^1 ball. This answers a question raised by Kozma (2007).
ER -