On the upper tail large deviation rate function for chemical distance in supercritical percolation

Abstract

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 $\mu (x)$ such that the chemical distance $\mathcal{D}(0,nx)$ between two connected points $0$ and $nx$ grows like $n\mu (x)$. Garet and Marchand (Ann. Prob., 2007) proved that the probability of the upper tail large deviation event $\left\{n\mu (x)(1+\epsilon )<\mathcal{D}(0,nx)<\infty \right\}$ 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 $\epsilon >0$ is small enough. Moreover, we show that for any $\epsilon >0$, 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.

Extended introduction, small mistakes and typos have been fixed