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Positive association of the oriented percolation cluster in randomly oriented graphs

  • François Bienvenu (a1) (a2)

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 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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References

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[1]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 183185.
[2]Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation, Oxford Studies in Probability, Oxford University Press.
[3]Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12 9991040.
[4]Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38 14661474.
[5]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 593629.
[6]Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22 89103.
[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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