Percolation and first-passage percolation on oriented graphs

Olivier Garet (1) , Régine Marchand (1)

1 IECL - Institut Élie Cartan de Lorraine

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

Mots clés: sharp transition.; first-passage percolation; percolation

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Percolation and first-passage percolation on oriented graphs. Electronic Communications in Probability, 2021, 26, paper 50. ⟨10.1214/21-ECP419⟩. ⟨hal-01837212v2⟩