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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089270

Digital Object Identifier
doi:10.1214/14-AIHP639

Mathematical Reviews number (MathSciNet)
MR3449304

Zentralblatt MATH identifier
1336.60184

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aihp/1452089270


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References

  • [1] D. J. Aldous. Markov chains with almost exponential hitting times. Stochastic Process. Appl. 13 (1982) 305–310.
  • [2] D. J. Aldous and J. A. Fill. Reversible Markov Chains and Random walks on Graphs. Unfinished monograph, recompiled 2014. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [3] P. Caputo, T. M. Liggett and T. Richthammer. Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 (2010) 831–851.
  • [4] Y.-T. Chen. Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. Ann. Appl. Probab. 23 (2013) 637–664.
  • [5] J. T. Cox. Coalescing random walks and voter model consensus times on the torus in $\mathbb{Z}^{d}$. Ann. Probab. 17 (1989) 1333–1366.
  • [6] J. T. Cox. Intermediate range migration in the two-dimensional stepping stone model. Ann. Appl. Probab. 20 (2010) 785–205.
  • [7] J. T. Cox, R. Durrett and E. A. Perkins. Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28 (2000) 185–234.
  • [8] J. T. Cox, R. Durrett and E. A. Perkins. Voter model perturbations and reaction diffusion equations. Astérisque 349 (2013) 1–113.
  • [9] J. T. Cox, M. Merle and E. A. Perkins. Coexistence in a two-dimensional Lotka–Volterra model. Electron. J. Probab. 15 (2010) 1190–1266.
  • [10] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695–750.
  • [11] R. Durrett. Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 2010.
  • [12] P. Donnelly and D. Welsh. Finite particle systems and infection models. Math. Proc. Cambridge Philos. Soc. 94 (1983) 167–182.
  • [13] S. N. Ethier and T. G. Kurtz. Markov Processes, Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, 1986.
  • [14] J. Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195 (910) (2008) 1–100.
  • [15] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003.
  • [16] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications. Springer, New York, 2002.
  • [17] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009.
  • [18] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985.
  • [19] T. M. Liggett. Continuous Time Markov Processes. An Introduction. Graduate Studies in Mathematics 113. Amer. Math. Soc., Providence, RI, 2010.
  • [20] L. Lovász. Random walks on graphs: A survey. In Combinatorics, Paul Erdös Is Eighty, Vol. 2 (Keszthely, 1993). Bolyai Soc. Math. Stud. 2 353–397. János Bolyai Math. Soc., Budapest, 1996.
  • [21] E. Lubetzky and A. Sly. Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153 (2010) 475–510.
  • [22] C. Mueller and R. Tribe. Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields 102 (1995) 519–545.
  • [23] R. I. Oliveira. Mean field conditions for coalescing random walks. Ann. Probab. 41 (2013) 3420–3461.
  • [24] R. I. Oliveira. On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364 (2012) 2109–2128.
  • [25] H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441 (2006) 502–505.
  • [26] V. Sood and S. Redner. Voter model on heterogeneous graphs. Phys. Rev. Lett. 94 (2005) 178701.
  • [27] S. Tavaré. Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 (1984) 119–164.