发表机构
School of Mathematics, Monash University（蒙纳士大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了整数格上受激随机游走访问范围在适当缩放后依概率收敛到显式ℓ1球，解决了Kozma提出的问题。
AI 中文摘要
我们研究了在 $\mathbb Z^d$（$d\geq2$）上受激向中心随机游走其值域的极限形状。在每个非零顶点处放置一块饼干。当游走者首次访问此类顶点时，它会消耗饼干，并以其下一步最近邻步长偏向原点。当访问原点或后续访问非零顶点时，没有饼干可用，游走者均匀随机地移动到邻居。我们证明了截至时间 $n$ 访问的顶点集合，按 $n^{-1/(d+1)}$ 缩放后，在豪斯多夫距离下依概率收敛到一个显式的 $\ell^1$ 球。这回答了 Kozma（2007）提出的一个问题。
英文摘要
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).
Comments31 pages, 1 figure
