Consider any fixed graph whose edges have been randomly and independently oriented, and write \(\{S \leadsto i\}\) 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 \(j\), \(\{S \leadsto i\}\) and \(\{S \leadsto j\}\) 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 \(\{S \leadsto i\}\), \(i \in V\), are positively associated -- meaning that the expectation of the product of increasing functionals of the family \(\{S \leadsto i\}\) for \(i \in V\) is greater than the product of their expectations.
To show how this result can be used in concrete calculations, I also detail the example of percolation from the leaves of the randomly oriented complete binary tree of height \(n\). Positive association makes it possible to use the Stein--Chen method to find conditions for the size of the percolation cluster to be Poissonian in the limit as \(n\) goes to infinity.