论文标题
涵盖了晶体晶格上第一段渗透的极限形状的单调性
Covering monotonicity of the limit shapes of first passage percolation on crystal lattices
论文作者
论文摘要
本文研究了第一个通道渗透（FPP）模型：分配了立方晶格中的每个边缘一个随机通道时间，并考虑了渗透率区域$ b（t）$的行为，该行为由可以在$ t> 0 $ $ t> 0 $内从原点到达的这些顶点组成。考克斯和杜雷特（Cox）和杜雷特（Durrett）展示了渗透区域的形状定理，称归一化的区域$ b（t）/t $收敛到一定极限形状$ \ mathcal {b} $。本文介绍了在晶体晶格上定义的一般FPP模型，并显示了覆盖地图下极限形状的单调性，从而提供了对立方FPP模型的极限形状的见解。
This paper studies the first passage percolation (FPP) model: each edge in the cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This paper introduces a general FPP model defined on crystal lattices, and shows the monotonicity of the limit shapes under covering maps, thereby providing insight into the limit shape of the cubic FPP model.
