Positive association of the oriented percolation cluster in randomly oriented graphs

Abstract

Consider any fixed graph whose edges have been randomly and independently oriented, and write $\{S⇝i\}$ to indicate that there is an oriented path going from a vertex $s\in S$ to vertex $i$. Narayanan (2016) proved that for any set $S$ and any two vertices $i$ and $j$, $\{S⇝i\}$ and $\{S⇝j\}$ are positively correlated. His proof relies on the Ahlswede-Daykin inequality, a rather advanced tool of probabilistic combinatorics. In this short note, I give an elementary proof of the following, stronger result: writing $V$ for the vertex set of the graph, for any source set $S$, the events $\{S⇝i\}$, $i\in V$, are positively associated -- meaning that the expectation of the product of increasing functionals of the family $\{S⇝i\}$ for $i\in V$ is greater than the product of their expectations.

This is the accepted (post-print) version. The example of application was removed