平衡激发随机游走 M(2,1,2) 的常返性与访问范围

中文摘要 AI 辅助

我们证明了平面平衡激发随机游走 $M(2,1,2)$ 是常返的。该游走在首次离开每个顶点时采取水平简单随机游走步，在之后每次离开时采取平面简单随机游走步。此外，在时间 $n$ 之前访问的不同顶点数乘以 $(\log n)/n$ 几乎必然收敛于 $\pi$，并且在每个 $L^p$（$1\le p<\infty$）中收敛，该极限与平面简单随机游走相同。更一般地，我们证明了在空间非均匀的饼干环境中，当每个顶点的总正、负饼干强度分别被常数 $A, B$ 所界且满足 $A+B<1+1/(2\pi +1)$ 时，平衡激发随机游走是常返的。

英文摘要

We prove that the planar balanced excited random walk $M(2,1,2)$ is recurrent. This walk takes a horizontal simple random walk step on its first departure from each vertex and a planar simple random walk step on every later departure. Moreover, the number of distinct vertices visited before time $n$, multiplied by $(\log n)/n$, converges to $π$ almost surely and in every $L^p$, $1\le p<\infty$, the same limit as for the planar simple random walk. More generally, we prove recurrence of balanced excited random walks in spatially inhomogeneous cookie environments whenever the total positive and negative cookie strengths at each vertex are bounded by constants $A, B$ with $A+B<1+1/(2π+1)$.

发表机构

• Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）
• Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）

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