论文标题
比较Bernoulli第一页渗透的极限形状
Comparison of limit shapes for Bernoulli first-passage percolation
论文作者
论文摘要
我们认为$ d $ d $ d $ d \ geq 2 $的Bernoulli首次渗透率。 Edge $ e $的通行时间为$ 0 $，概率$ p $，$ 1 $，概率$ 1-p $，彼此独立。令$ p_c $是带有通道时间$ 0 $的边缘渗透的关键概率。当$ 0 \ leq p <p_c $时，存在非随机，非空的紧凑型凸面集$ \ nathcal {b} _p $，以至于从$ t $中$ t $ t \ nathcal {b} _p $ for probibity oppartice oppartice opprobine oppropy opploximition optike opart $ t $ oppains opt $ t $ for probibity for y MathCal {本文的目的是证明$ 0 \ leq p <q <q <p_c $，$ \ mathcal {b} _p $和$ \ mathcal {b} _q $之间的hausdorff距离在$ q-p $中线性增长。此外，我们提到的是，本文采用的方法为地理学交汇处的预期大小提供了一个下限，这给\ textit {critical {critical}案例带来了非平凡的后果。
We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. The passage time of edge $e$ is $0$ with probability $p$ and $1$ with probability $1-p$, independently of each other. Let $p_c$ be the critical probability for percolation of edges with passage time $0$. When $0\leq p<p_c$, there exists a nonrandom, nonempty compact convex set $\mathcal{B}_p$ such that the set of vertices to which the first-passage time from the origin is within $t$ is well-approximated by $t\mathcal{B}_p$ for all large $t$, with probability one. The aim of this paper is to prove that for $0\leq p<q<p_c$, the Hausdorff distance between $\mathcal{B}_p$ and $\mathcal{B}_q$ grows linearly in $q-p$. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the \textit{critical} case.
