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: E $\mathrm{\ell }$ = {x $\in$ R 2 |f (x) $\ge$-$\mathrm{\ell }$}, where $\mathrm{\ell }$ $\in$ R is a real parameter and f is a continuous, stationary, centered, planar Gaussian field satisfying some regularity 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 $\mathrm{\ell }$c = 0. When $\mathrm{\ell }$ > 0, we are in a supercritical regime and almost surely E $\mathrm{\ell }$ has a unique unbounded connected component. We prove that in this supercritical regime, whenever two points are in the same connected components of E $\mathrm{\ell }$ then, with high probability, the chemical distance (the length of the shortest path in E $\mathrm{\ell }$ between these points) is close to the Euclidean distance between those two points Contents