Positive association of the oriented percolation cluster in randomly oriented graphs

François Bienvenu

Published 2017-11-23Version 1

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.