Variance bounds for Gaussian first passage percolation

Abstract

Recently, many results have been established drawing a parallel between Bernoulli percolation and models given by levels of smooth Gaussian fields with unbounded, strongly decaying correlation (see e.g [2], [12], [10]). In a previous work with D. Gayet [5], we started to extend these analogies by adapting the first basic results of classical first passage percolation (first established in [8], [4]) in this new framework: positivity of the time constant and the ball-shape theorem. In the present paper, we present a proof inspired by [9] of other basic properties of the new FPP model: an upper bound on the variance in the FPP pseudometric given by the Euclidean distance with a logarithmic factor, and a constant lower bound. Our results notably apply to the Bargmann-Fock field.