A note on correlations in randomly oriented graphs

Given a graph $G$, we consider the model where $G$ is given a random orientation by giving each edge a random direction. It is proven that for $a,b,s\in V(G)$, the events $\{s\to a\}$ and $\{s\to b\}$ are positively correlated. This correlation persists, perhaps unexpectedly, also if we first condition on \{s\nto t\} for any vertex $t\neq s$. With this conditioning it is also true that $\{s\to b\}$ and $\{a\to t\}$ are negatively correlated. A concept of increasing events in random orientations is defined and a general inequality corresponding to Harris inequality is given. The results are obtained by combining a very useful lemma by Colin McDiarmid which relates random orientations with edge percolation, with results by van den Berg, H\"aggstr\"om, Kahn on correlation inequalities for edge percolation. The results are true also for another model of randomly directed graphs.Comment: 7 pages. The main lemma was first published by Colin McDiarmid. Relevant reference added and text rewritten to reflect this fac

On percolation and the bunkbed conjecture

We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent of other edges. The Bunkbed conjecture states that for all $G$ and $p$ the probability that $(u,0)$ is in the same component as $(v,0)$ is greater than or equal to the probability that $(u,0)$ is in the same component as $(v,1)$ for every pair of vertices $u,v\in G$. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs $G$, in particular outerplanar graphs.Comment: 13 pages, improved exposition thanks to anonymous referee. To appear in CP