Stationary distributions for the voter model in d≥3 are factors of IID

Allan Sly; Lingfu Zhang

doi:10.1214/22-AOP1568

Abstract

For the Voter Model on ${\mathbb{Z}^{d}}$ , $d\ge 3$ , 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 ${\mathbb{Z}^{d}}$ .

Funding Statement

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

Acknowledgments

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 ${\mathfrak{W}_{0}}$ and ${\mathfrak{W}_{1}}$ in Section 2.2.

Citation

Download Citation

Allan Sly. Lingfu Zhang. "Stationary distributions for the voter model in $d\ge 3$ are factors of IID." Ann. Probab. 50 (4) 1589 - 1609, July 2022. https://doi.org/10.1214/22-AOP1568

Information

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

Subjects:

Primary: 37A50 , 82C22

Secondary: 37A35 , 60G10 , 60K35 , 82B20

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

Abstract

Funding Statement

Acknowledgments

Citation

Information

KEYWORDS/PHRASES

PUBLICATION TITLE:

PUBLICATION YEARS