Random pseudometrics and applications

Let $T$ be a random ergodic pseudometric over $\mathbb R^d$. This setting generalizes the classical \emph{first passage percolation} (FPP) over $\mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-independence, that ensure the positivity of its time constants, that is almost surely, the pseudo-distance given by $T$ from the origin is asymptotically a norm. Combining this general result with previously known ones, we prove that The known phase transition for Gaussian percolation in the case of fields with positive correlations with exponentially fast decayholds for Gaussian FPP, including the natural Bargmann-Fock model; The known phase transition for Voronoi percolation also extends to the associated FPP; The same happens for Boolean percolation for radii with exponential tails, a result which was known without this condition. We prove the positivity of the constant for random continuous Riemannian metrics, including cases with infinite correlations in dimension $d=2$. Finally, we show that the critical exponent for the one-arm, if exists, is bounded above by $d-1$. This holds forbond Bernoulli percolation, planar Gaussian fields, planar Voronoi percolation, and Boolean percolation with exponential small tails.