Positive association of the oriented percolation cluster in randomly oriented graphs

TL;DR

This paper provides an elementary proof that in randomly oriented graphs, the events of reaching different vertices from a source set are positively associated, strengthening previous results that used complex probabilistic inequalities.

Contribution

It offers a simpler, more accessible proof of positive association in oriented percolation clusters, improving upon prior complex methods.

Findings

Events $ ext{S} o i$ are positively associated for all vertices $i$.

Positive association holds for increasing functionals of reachability events.

Elementary proof replaces advanced probabilistic inequalities.

Abstract

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.