Limit shapes for centrally excited walks

Abstract

A centrally excited random walk on ${\mathbb{Z}}^d$ 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 $d\ge 2$ 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.

arXiv comment29 pages, 5 figures. Lean formalization: https://github.com/nitromannitol/Centrally-Excited-Random-Walk