July 2022 Stationary distributions for the voter model in d3 are factors of IID
Allan Sly, Lingfu Zhang
Author Affiliations +
Ann. Probab. 50(4): 1589-1609 (July 2022). DOI: 10.1214/22-AOP1568


For the Voter Model on Zd, d3, we show that the (extremal) stationary distributions are isomorphic to Bernoulli shifts, and answer an open question asked by Steif and Tykesson in (ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019) 899–955). The proof gives explicit constructions of the stationary distributions as factors of IID processes on Zd.

Funding Statement

This work was supported by an NSF career award (DMS-1352013), a Simons Investigator grant and a MacArthur Fellowship.


The authors would like to thank Jeff Steif for suggesting the problem. We are also very thankful for the many helpful suggestions from an anonymous referee, who pointed out an error in the first version of this paper, and suggested a simplification in the construction of W0 and W1 in Section 2.2.


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Allan Sly. Lingfu Zhang. "Stationary distributions for the voter model in d3 are factors of IID." Ann. Probab. 50 (4) 1589 - 1609, July 2022. https://doi.org/10.1214/22-AOP1568


Received: 1 March 2021; Revised: 1 January 2022; Published: July 2022
First available in Project Euclid: 11 May 2022

MathSciNet: MR4420427
zbMATH: 1498.37085
Digital Object Identifier: 10.1214/22-AOP1568

Primary: 37A50 , 82C22
Secondary: 37A35 , 60G10 , 60K35 , 82B20

Keywords: Bernoulli shift , factor of iid , random cluster model , voter model

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 4 • July 2022
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