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

We consider supercritical bond percolation on $\mathbb Z^d$ and study the chemical distance, i.e., the graph distance on the infinite cluster. 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) prove that the probability of the upper tail large deviation event $\left\{\mathcal {D}(0,nx)>n\mu(x)(1+\epsilon), 0\leftrightarrow nx\right\}$ decays exponentially in $n$. In this paper, we prove the existence of the rate function for the upper tail large deviation when $d\ge 3$ and $\epsilon>0$ is small enough. We prove that the upper tail large deviation event is created by a space-time cut-point (a point that any geodesic from $0$ to $nx$ must cross after a given time) that forces the geodesics to "loose time" by going in a non-optimal direction or by wiggling a lot. This enables us to express the rate function in terms of the rate function for a space-time cut-point