Abstract
We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph of initially occupied edges on n vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is . We conclude with several conjectures and open problems.
Funding Statement
J. Gravner was partially supported by NSF Grant DMS-1513340 and the Slovenian Research Agency research program P1-0285. B. Kolesnik was partially supported supported by an NSERC Postdoctoral Fellowship.
Acknowledgments
We thank the referees for their comments that helped improve this work.
Citation
Janko Gravner. Brett Kolesnik. "Transitive closure in a polluted environment." Ann. Appl. Probab. 33 (1) 107 - 126, February 2023. https://doi.org/10.1214/22-AAP1810
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