Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the convergence of densities of finite voter models to the Wright–Fisher diffusion

Yu-Ting Chen, Jihyeok Choi, and J. Theodore Cox

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We study voter models defined on large finite sets. Through a perspective emphasizing the martingale property of voter density processes, we prove that in general, their convergence to the Wright–Fisher diffusion only involves certain averages of the voter models over a small number of spatial locations. This enables us to identify suitable mixing conditions on the underlying voting kernels, one of which may just depend on their eigenvalues in some contexts, to obtain the convergence of density processes. We show by examples that these conditions are satisfied by a large class of voter models on growing finite graphs.


Nous étudions le modèle du votant sur des ensembles contenant un nombre grand mais fini de sites. En nous servant des propriétés de martingales des densités du modèle du votant nous prouvons qu’ il y a convergence vers une diffusion Wright–Fisher. De plus cette preuve de convergence n’utilise que certaines moyennes sur un petit nombre de sites. Ceci nous permet d’identifier des conditions de mélange concernant le noyau du votant sous-jacent. Dans certains cas une de ces conditions nous permet de démontrer la convergence des densités en n’utilisant que les valeurs propres des noyaux. Nous donnons des exemples montrant que ces conditions de mélange sont satisfaites pour une grande classe de modèles du votant sur des ensembles croissants de graphes.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 286-322.

Received: 22 November 2013
Revised: 22 July 2014
Accepted: 4 August 2014
First available in Project Euclid: 6 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]
Secondary: 60F05: Central limit and other weak theorems 60J60: Diffusion processes [See also 58J65]

Wright–Fisher diffusion Voter model Interacting particle system Dual processes Semimartingale convergence theorem


Chen, Yu-Ting; Choi, Jihyeok; Cox, J. Theodore. On the convergence of densities of finite voter models to the Wright–Fisher diffusion. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 286--322. doi:10.1214/14-AIHP639.

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