发表机构
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究确定了二维离散环面上简单随机游走覆盖时间的极限分布，证明其归一化后收敛于Gumbel变量与临界高斯乘法混沌总质量对数之和，回答了Aldous提出的极限定律问题。
AI 中文摘要
我们确定了二维离散环面上简单随机游走覆盖时间的极限分布。对于总跳跃率为1的连续时间游走在$(\mathbb Z/N\mathbb Z)^2$上，令$T_N$表示其覆盖时间。我们证明$\frac{T_N}{(2/\pi)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\log(\kappa Z)$，其中$G$是标准Gumbel随机变量，$Z$是单位环面上零平均高斯自由场相关的临界高斯乘法混沌的总质量，$G$和$Z$独立，且$\kappa>0$是确定性的。这回答了Aldous提出并由Dembo、Peres、Rosen和Zeitouni记录的极限定律问题。证明识别了在覆盖前一个确定时刻，小而分离的未访问组件的数量中的随机波动，然后估计访问剩余组件所需的时间。
英文摘要
We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\mathbb Z/N\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\frac{T_N}{(2/π)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\log(κZ)$, where $G$ is a standard Gumbel random variable, $Z$ is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, $G$ and $Z$ are independent, and $κ>0$ is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.
Comments203 pages
