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: percolation; first-passage percolation; sharp transition.

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Percolation and first-passage percolation on oriented graphs. 2018. ⟨hal-01837212v1⟩