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Published November 2021 | Accepted Version
Journal Article Open

No Percolation at Criticality on Certain Groups of Intermediate Growth


We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form p_n(v,v) ≤ exp[−Ω(n^γ)] for some γ > ½⁠. The result is new in the case that the graph is of intermediate volume growth.

Additional Information

© The Author(s) 2020. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Received: 01 October 2018; Revision received: 06 September 2019; Accepted: 10 September 2019; Published: 06 December 2019. We thank Gidi Amir and Tianyi Zheng for sharing their expertise on groups of intermediate growth. This work was supported by Engineering and Physical Sciences Research Council (EPSRC) (EP/L018896/1 to J.H.).

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August 20, 2023
October 23, 2023