arXiv 2104.012112021-04math.PRmath-phmath.MP

二维近临界首过渗流极限形状的收敛性

Convergence of limit shapes for 2D near-critical first-passage percolation

Chang-Long Yao

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

摘要：

我们考虑三角形格子上的伯努利首次通过渗流，其中站点具有0和1的通过时间概率分别为$p$和$1-p$。对于每个$p\in(0,p_c)$，令$\mathcal {B}(p)$为经典“形状定理”中的极限形状，并令$L(p)$为相关长度。我们发现，作为$p\uparrow p_c$，重新缩放的极限形状$L(p)^{-1}\mathcal {B}(p)$收敛到一个欧几里得磁盘。这改进了Chayes等人[J. Stat. Phys.45（1986）933- 951]。该证明依赖于Garban等人[J. Eur. Math.Soc.20（2018）1195- 1268]，并使用Camia等人引入的标度极限中的连续体簇集合的构造[Springer Proceedings in Mathematics \& Statistics，299（2019）44- 89]。

英文摘要：

We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the classical "shape theorem", and let $L(p)$ be the correlation length. We show that as $p\uparrow p_c$, the rescaled limit shape $L(p)^{-1}\mathcal {B}(p)$ converges to a Euclidean disk. This improves a result of Chayes et al. [J. Stat. Phys. 45 (1986) 933--951]. The proof relies on the scaling limit of near-critical percolation established by Garban et al. [J. Eur. Math. Soc. 20 (2018) 1195--1268], and uses the construction of the collection of continuum clusters in the scaling limit introduced by Camia et al. [Springer Proceedings in Mathematics \& Statistics, 299 (2019) 44--89].

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摘要：

英文摘要：

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