Limit shapes for centrally excited walks
Abstract
A centrally excited random walk on ℤd moves like simple random walk, except that its first step from each site other than the origin has a drift of fixed size toward it. Kozma (2007) conjectured that after n steps the visited set approximates a ball with radius of order n1/(d+1). We prove this in every dimension d≥ 2 and show that the rescaled local times converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that, when the drift is opposite to a subgradient of a norm, the visited set approximates the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most n1/6 times a power of log n, and the exponent 1/6 cannot be lowered. Earlier versions of this work appeared in three independent drafts by Nguyen (arXiv:2610.07724), Bou-Rabee and Peres (arxiv:2610.10162v1) and Singh (arxiv:2610.12138). The present paper brings together the results and ideas from these drafts.
Related
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- Functional limit theorems for perturbed random walks
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BibTeX
@misc{bourabee2026limit,
title = {{Limit shapes for centrally excited walks}},
author = {Ahmed Bou-Rabee and Tuan-Minh Nguyen and Yuval Peres and Arvind Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10162},
eprint = {2610.10162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10162},
}
RIS
TY - UNPB
TI - Limit shapes for centrally excited walks
AU - Ahmed Bou-Rabee
AU - Tuan-Minh Nguyen
AU - Yuval Peres
AU - Arvind Singh
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.10162
UR - https://arxiv.org/abs/2610.10162
AB - A centrally excited random walk on ℤ^d moves like simple random walk, except that its first step from each site other than the origin has a drift of fixed size toward it. Kozma (2007) conjectured that after n steps the visited set approximates a ball with radius of order n^1/(d+1). We prove this in every dimension d≥ 2 and show that the rescaled local times converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that, when the drift is opposite to a subgradient of a norm, the visited set approximates the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most n^1/6 times a power of log n, and the exponent 1/6 cannot be lowered. Earlier versions of this work appeared in three independent drafts by Nguyen (arXiv:2610.07724), Bou-Rabee and Peres (arxiv:2610.10162v1) and Singh (arxiv:2610.12138). The present paper brings together the results and ideas from these drafts.
ER -