  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. In a previous work with D. Gayet , we
started to extend these analogies by adapting the first basic results of
classical first passage percolation in this new framework: positivity of the
time constant and the ball-shape theorem. In the present paper, we present a
proof inspired by Kesten 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.
