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Positive association of the oriented percolation cluster in randomly oriented graphs

Part of: Combinatorial probability Graph theory Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  08 July 2019

François Bienvenu
François Bienvenu*
Affiliation:
Center for Interdisciplinary Research in Biology (CIRB), CNRS UMR 7241, Collège de France, Paris, France Laboratoire de Probabilités, Statistique et Modélisation (LPSM), CNRS UMR 8001, Sorbonne Université, Paris, France
*
(Email: francois.bienvenu@normalesup.org)
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Abstract

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 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.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 811 - 815
Copyright
© Cambridge University Press 2019 

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