Shadow and percolation III: chemical distance in continuous landscapes with correlations

Abstract

We study some geometric properties of the excursion set of a slope field alpha associated to a smooth, planar, centered, Gaussian field f. That, is we consider the set of all points such that the value of alpha is at most l where l is a real parameter called the level. We restrict our attention to the levels l that are supercritical. We show that for almost such l, in the sense of the Lebesgue measure, then with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This result is in the spirit of the Antal Pisztora theorem for Bernoulli percolation. However, many new difficulties arise such as the fact that alpha is a continuous field (not differentiable everywhere) with long range correlations and whose law is still not well understood.