Chemical distance in the supercritical phase of planar Gaussian fields

Abstract

Our study concerns the large scale geometry of the excursion set of planar random fields: E ${\ell}$ = {x $\in$ R 2 |f (x) $\ge$-${\ell}$}, where ${\ell}$ $\in$ 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 ${\ell}$c = 0. When ${\ell}$ > 0, we are in a supercritical regime and almost surely E ${\ell}$ has a unique unbounded connected component. We prove that in this supercritical regime, whenever two points are in the same connected components of E ${\ell}$ then, with high probability, the chemical distance (the length of the shortest path in E ${\ell}$ between these points) is close to the Euclidean distance between those two points Contents