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Positive association of the oriented percolation cluster in randomly oriented graphs

  • François Bienvenu (a1) (a2)


Consider any fixed graph whose edges have been randomly and independently oriented, and write {S ⇝} to indicate that there is an oriented path going from a vertex sS to vertex i. Narayanan (2016) proved that for any set S and any two vertices i and j, {Si} and {Sj} 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 {Si}, iV, are positively associated, meaning that the expectation of the product of increasing functionals of the family {Si} for iV is greater than the product of their expectations.


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[7]Linusson, S. (2009) A note on correlations in randomly oriented graphs. arXiv:0905.2881
[8]Narayanan, B. (2018) Connections in randomly oriented graphs. Combin. Probab. Comput. 27 667671.
[9]Ross, N. (2011) Fundamentals of Stein’s method. Probab. Surv. 8 201293.

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Positive association of the oriented percolation cluster in randomly oriented graphs

  • François Bienvenu (a1) (a2)


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