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2021 The q-voter model on the torus
Pooja Agarwal, Mackenzie Simper, Rick Durrett
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Electron. J. Probab. 26: 1-33 (2021). DOI: 10.1214/21-EJP682


In the q-voter model, the voter at x changes its opinion at rate fxq, where fx is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if q<1 and clustering if q>1. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for q close to 1. More precisely, we show that if q<1, then for any m< the process on the three-dimensional torus with n points survives for time nm, and after an initial transient phase has a density that it is always close to 1/2. Readers familiar with long time survival results for the contact process and other praticle systems might expect the conjecture to say survival occurs for time exp(γn) with γ>0, however we show persistence does not hold for exp(nβ) with β>13. If q>1, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time logn but the stochastic process on the same time scale dies out at time (13)logn.

Funding Statement

PA was partially supported by the NSF Grant DMS 1713032. MS was supported by a National Defense Science & Engineering Graduate Fellowship. RD was partially supported by NSF grant DMS 1809967 from the probability program.


This work was begun during the 2019 AMS Math Research Communities meeting on Stochastic Spatial Models, June 9–15, 2019. We would like to thank Hwai-Ray Tung, a graduate student at Duke for producing the figures.


Download Citation

Pooja Agarwal. Mackenzie Simper. Rick Durrett. "The q-voter model on the torus." Electron. J. Probab. 26 1 - 33, 2021.


Received: 4 June 2020; Accepted: 6 August 2021; Published: 2021
First available in Project Euclid: 20 September 2021

Digital Object Identifier: 10.1214/21-EJP682

Primary: 60K35

Keywords: ODE limit , renormalization , voter model perturbation


Vol.26 • 2021
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