Percolation and first-passage percolation on oriented graphs

TL;DR

This paper explores percolation and first-passage percolation on oriented, translation-invariant graphs on , revealing direction-dependent critical probabilities and establishing the sharpness of phase transitions in specific directions.

Contribution

It provides the first analysis of Bernoulli percolation on oriented graphs with loops, demonstrating direction-dependent critical probabilities and linking percolation with first-passage percolation.

Findings

Critical probability can vary with direction.

Phase transition is sharp in a given direction.

Links established between percolation and first-passage percolation.

Abstract

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.