Variance bounds for Gaussian first passage percolation

TL;DR

This paper extends classical first passage percolation results to Gaussian fields, establishing variance bounds and shape theorems for models with correlated Gaussian levels, notably applying to the Bargmann-Fock field.

Contribution

It introduces variance bounds and fundamental properties for Gaussian FPP models, expanding the classical theory to correlated Gaussian fields.

Findings

Variance upper bound with logarithmic factor

Constant lower bound on variance

Application to Bargmann-Fock field

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. 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.