论文标题
在上尾上，超临界渗透中的化学距离的较大偏差率函数
On the upper tail large deviation rate function for chemical distance in supercritical percolation
论文作者
论文摘要
我们考虑$ \ mathbb z^d $上的超临界键渗透，并研究称为化学距离的渗透图上的图形距离。众所周知，存在确定性常量的$μ（x）$，因此化学距离$ \ MATHCAL D（0，nx）$在两个连接点$ 0 $ 0 $和$ nx $之间，例如$nμ（x）$。 Garet and Marchand（Ann。prob。，2007）证明了上尾大偏差事件的概率$ \ left \ {nμ（x）（1+ \ varepsilon）<\ Mathcal d（0，nx）<\ nx）<\ iffty \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \ right \} $相对于$ n $ n $ n $ $ n $。在本文中，当$ d \ ge 3 $和$ \ varepsilon> 0 $时，我们证明了上尾大偏差的速率功能的存在。此外，我们表明，对于任何$ \ varepsilon> 0 $，上尾大偏差事件都是由时空裁切点创建的（所有路径从$ 0 $ $ 0 $到$ nx $都必须在给定时间后都必须交叉），迫使大地测量学通过在非优势方向或摇动来消耗更多时间。这使我们能够在时空切点方面表达速率函数。
We consider the supercritical bond percolation on $\mathbb Z^d$ and study the graph distance on the percolation graph called the chemical distance. It is well-known that there exists a deterministic constant $μ(x)$ such that the chemical distance $\mathcal D(0,nx)$ between two connected points $0$ and $nx$ grows like $nμ(x)$. Garet and Marchand (Ann. Prob., 2007) proved that the probability of the upper tail large deviation event $\left\{nμ(x)(1+\varepsilon)<\mathcal D(0,nx)<\infty\right\} $ decays exponentially with respect to $n$. In this paper, we prove the existence of the rate function for upper tail large deviation when $d\ge 3$ and $\varepsilon>0$ is small enough. Moreover, we show that for any $\varepsilon>0$, the upper tail large deviation event is created by space-time cut-points (points that all paths from $0$ to $nx$ must cross after a given time) that force the geodesics to consume more time by going in a non-optimal direction or by wiggling considerably. This enables us to express the rate function in regards to space-time cut-points.
