Chemical distance for smooth Gaussian fields in higher dimension

Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on Z d .Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field f and we study the topological and geometric properties of the random excursion set E ℓ (f ) := {x ∈ R d | f (x) ≥ -ℓ} where ℓ ∈ R is called a level.It is known that for a wide variety of fields f , there exists a phase transition at some critical level ℓ c .When ℓ > ℓ c , the excursion set E ℓ (f ) presents a unique unbounded component while if ℓ < ℓ c there are only bounded components in E ℓ (f ).In the supercritical regime, ℓ > ℓ c , we study the geometry of the unbounded cluster.Inspired by the work of Antal and Pisztora (1996) for the Bernoulli model, we introduce the chemical distance between two points x and y as the Euclidean length of the shortest path connecting these points and staying in E ℓ (f ).In this paper, we show that when ℓ > -ℓ c then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points.