Percolation and first-passage percolation on oriented graphs

Abstract

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices ${\mathbb{Z}}^d$ 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.