# SMILE

## Positive association of the oriented percolation cluster in randomly oriented graphs

### 2019

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.

# Bibtex

@article{Bienvenu2017PercoRandomlyOriented,
author = {Bienvenu, Fran{\c{c}}ois},
title = {Positive association of the oriented percolation cluster in randomly oriented graphs},
journal = {Combinatorics, Probability and Computing},
volume = {28},
number = {6},
pages = {811--815}
year = {2019},
doi = {10.1017/S0963548319000191}
}