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Published September 2016 | Submitted
Journal Article Open

Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters


We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.

Additional Information

© 2016 Académie des sciences. Published by Elsevier Masson SAS Under an Elsevier user license. Received 3 July 2016, Accepted 27 July 2016, Available online 8 August 2016. This work was supported by a Microsoft Research PhD Fellowship and was carried out while the author was an intern at Microsoft Research, Redmond. We thank Ander Holroyd, Gady Kozma, Russ Lyons, Asaf Nachmias, and Tianyi Zheng for helpful comments and suggestions. We also thank Hugo Duminil-Copin and Ben Wallace for the french translation of the title and abstract.

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