Random pseudometrics and applications

Vivek Dewan (1) , Damien Gayet (2)

1 ENS de Lyon - École normale supérieure de Lyon
2 IF - Institut Fourier

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 decay holds 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 for bond Bernoulli percolation, planar Gaussian fields, planar Voronoi percolation, and Boolean percolation with exponential small tails.

Domaines: Probabilités [math.PR]

Vivek Dewan, Damien Gayet. Random pseudometrics and applications. 2020. ⟨hal-02539503⟩