Percolation on Bidirected Graphs

Abstract

We study bidirected percolation on an undirected graph $G$: each edge is replaced by two oppositely directed edges, each of which is open independently with probability $p$. Two vertices are strongly connected if they are mutually reachable by open directed paths. Let $p_{\mathrm{sc}}$ denote the threshold for the existence of an infinite strongly connected cluster, and let $p_c$ denote the usual Bernoulli percolation threshold on $G$. On every transitive amenable graph, we prove that $p_{\mathrm{sc}}=p_c$. In contrast, sufficiently spread-out modifications of every transitive non-amenable graph satisfy $p_{\mathrm{sc}}>p_c$. We also show that, on ${\mathbb{Z}}^d$ with $d\ge 11$, the expected size of the strongly connected cluster is finite at $p=p_{\mathrm{sc}}$.

arXiv comment19 pages, 2 figures