Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-09T22:44:18.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive association of the oriented percolation cluster in randomly oriented graphs

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

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 811 - 815
Copyright
© Cambridge University Press 2019 

References

Ahlswede, R. and Daykin, D. E. (1978) An inequality for the weights of two families of sets, their unions and intersections. Probab. Theory Related Fields 43 183–185.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation, Oxford Studies in Probability, Oxford University Press.Google Scholar
Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12 999–1040.CrossRefGoogle Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38 1466–1474.CrossRefGoogle Scholar
Fill, J. A. and Pemantle, R. (1993) Percolation, first-passage percolation and covering times for Richardson’s model on the n-cube. Ann. Appl. Probab. 3 593–629.CrossRefGoogle Scholar
Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22 89–103.CrossRefGoogle Scholar
Linusson, S. (2009) A note on correlations in randomly oriented graphs. arXiv:0905.2881Google Scholar
Narayanan, B. (2018) Connections in randomly oriented graphs. Combin. Probab. Comput. 27 667–671.CrossRefGoogle Scholar
Ross, N. (2011) Fundamentals of Stein’s method. Probab. Surv. 8 201–293.CrossRefGoogle Scholar