Chemical distance in the supercritical phase of planar Gaussian fields

Our study concerns the large scale geometry of the excursion set of planar random fields: Eℓ={x∈R2|f(x)≥−ℓ}, where ℓ∈R is a real parameter and f is a continuous, stationary, centered, planar Gaussian field satisfying some regularity and decorrelation assumptions (in particular, this study applies to the planar Bargmann-Fock field). It is already known that under those hypotheses there is a phase transition at ℓc=0. When ℓ>0, we are in the supercritical regime and almost surely Eℓ has a unique unbounded connected component. We prove that in this supercritical regime, whenever two points are in the same connected components of Eℓ then, with high probability, the chemical distance (the length of the shortest path in Eℓ between these points) is comparable to the Euclidean distance between those two points up to some multiplicative constant.