AI 中文总结
该研究针对d维格点停车模型，在临界与亚临界状态下确定了访问次数、停车时间尾部的标度规律，采用Levine-Perès可分沙堆结合鞅项的方法，解答了Damron等人2019年提出的问题。
AI 中文摘要
在停车模型中，d维格点的每个位置以概率p独立初始化为一辆车，或以概率1-p独立初始化为一个停车位。车辆按独立的离散时间简单随机游走移动，并在找到第一个空闲停车位时停车。我们证明，在临界状态p=1/2下，对于d≤3，n轮内访问某位置的期望次数量级为n^((4-d)/4)；对于d≥4，该量级为log n。我们还证明，在亚临界状态p∈(0,1/2)下，停车时间的尾部由指数为d/(d+2)的 stretched exponentials（ stretched指数分布）上下界限定。当p趋近于1/2时，我们还确定了访问某位置的期望总次数的发散行为：在一维、二维、三维中，其量级分别为(1-2p)^{-3}、(1-2p)^{-1}、(1-2p)^{-1/3}；在四维及更高维中，其量级为log(1/(1-2p))。我们的证明采用了Levine和Peres提出的可分沙堆表示，加上一个鞅型项。这些结果回答了Damron、Gravner、Junge、Lyu和Sivakoff（2019）提出的问题。
英文摘要
In the parking model, each site of the $d$-dimensional lattice independently starts with one car with probability $p$ or one parking spot with probability $1-p$. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime $p=1/2$, the expected number of visits to a site in $n$ rounds is of order $n^{(4-d)/4}$ for $d\leq3$ and $\log n$ for $d\geq4$. We also prove that in the subcritical regime $p\in(0,1/2)$, the parking-time tail is bounded above and below by stretched exponentials with exponent $d/(d+2)$. As $p\uparrow1/2$, we also determine the divergence of the expected total number of visits to a site: its order is $(1-2p)^{-3}$, $(1-2p)^{-1}$ and $(1-2p)^{-1/3}$ in dimensions one, two and three, respectively, and $\log(1/(1-2p))$ in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).
Comments43 pages, 2 figures
