Positive association of the oriented percolation cluster in randomly oriented graphs

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 ∈ 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 ∈ V , are positively associated – meaning that the expectation of the product of increasing functionals of the family {S ⇝ i} for i ∈ 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.