二维近临界首达渗流极限形状的收敛性
概率论
2022-09-01 v3 数学物理
math.MP
摘要
我们考虑三角晶格上的 Bernoulli 首达渗流，其中位点具有 0 和 1 的通行时间，概率分别为 和 。对每个 ，令 为经典“形状定理”中的极限形状，并令 为关联长度。我们证明当 时，重标度极限形状 收敛到欧氏圆盘。这改进了 Chayes 等人 [J. Stat. Phys. 45 (1986) 933--951] 的结果。证明依赖于 Garban 等人 [J. Eur. Math. Soc. 20 (2018) 1195--1268] 建立的近临界渗流的标度极限，并使用了 Camia 等人 [Springer Proceedings in Mathematics \& Statistics, 299 (2019) 44--89] 引入的标度极限中连续统簇族构造。
引用
@article{arxiv.2104.01211,
title = {Convergence of limit shapes for 2D near-critical first-passage percolation},
author = {Chang-Long Yao},
journal= {arXiv preprint arXiv:2104.01211},
year = {2022}
}
备注
37 pages, 9 figures. The manuscript has been fully revised by following a referee's suggestions. Two main revisions to the previous version: 1) Section 3.3 is removed and the strip results are not used anymore. 2) $\mathcal {C}(z)$ is replaced by the cluster in $\mathbb{D}_{1/2}(z)$ with the largest diameter, and Section 4.2 is updated accordingly