David Vernotte

Given a planar continuous Gaussian field $f$, one can consider the percolation model associated with it. That is, define the excursion set as the random set of points in the plane such that the value of the field $f$ at this point is lower than some real level. It has been proved that this percolation model is very close to the Bernoulli percolation, in particular there is a phase transition at level $0$ (when $f$ is centered). When the level is striclty positive the excursion set presents a unique unbouded component (supercritical phase). In order to study the geometry of this component we introduce the chemical distance that measures the Euclidean length of the shortest path between two points that stays in the excursion set. By analogy with the Bernoulli percolation model, we prove that this chemical distance behaves with high probability like the Euclidean distance up to a small error.