The Annals of Probability

Percolation on the stationary distributions of the voter model

Balázs Ráth and Daniel Valesin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The voter model on $\mathbb{Z}^{d}$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d\geq3$, the set of (extremal) stationary distributions is a family of measures $\mu_{\alpha}$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_{\alpha}$ is a strongly correlated field of 0’s and 1’s on $\mathbb{Z}^{d}$ in which the density of 1’s is $\alpha$. We consider such a configuration as a site percolation model on $\mathbb{Z}^{d}$. We prove that if $d\geq5$, the probability of existence of an infinite percolation cluster of 1’s exhibits a phase transition in $\alpha$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d\geq3$.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1899-1951.

Dates
Received: February 2015
Revised: February 2016
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835234

Digital Object Identifier
doi:10.1214/16-AOP1104

Mathematical Reviews number (MathSciNet)
MR3650418

Zentralblatt MATH identifier
06754789

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 82B43: Percolation [See also 60K35]

Keywords
Interacting particle systems voter model percolation

Citation

Ráth, Balázs; Valesin, Daniel. Percolation on the stationary distributions of the voter model. Ann. Probab. 45 (2017), no. 3, 1899--1951. doi:10.1214/16-AOP1104. https://projecteuclid.org/euclid.aop/1494835234


Export citation

References

  • [1] Arratia, R. (1983). Site recurrence for annihilating random walks on $\textbf{Z}_{d}$. Ann. Probab. 11 706–713.
    Zentralblatt MATH: 0517.60075
    Digital Object Identifier: doi:10.1214/aop/1176993515
  • [2] Bramson, M., Cox, J. T. and Le Gall, J.-F. (2001). Super-Brownian limits of voter model clusters. Ann. Probab. 29 1001–1032.
    Zentralblatt MATH: 1029.60078
  • [3] Bramson, M. and Griffeath, D. (1980). Asymptotics for interacting particle systems on $\textbf{Z}^{d}$. Ann. Probab. 7 418–432.
    Zentralblatt MATH: 0417.60097
    Digital Object Identifier: doi:10.1007/BF01013315
  • [4] Bricmont, J., Lebowitz, J. L. and Maes, C. (1987). Percolation in strongly correlated systems: The massless Gaussian field. J. Stat. Phys. 48 1249–1268.
    Zentralblatt MATH: 0962.82520
    Digital Object Identifier: doi:10.1007/BF01009544
  • [5] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
    Zentralblatt MATH: 0272.60072
    Digital Object Identifier: doi:10.1093/biomet/60.3.581
  • [6] Cox, J. T., Durrett, R. and Perkins, E. A. (2000). Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28 185–234.
    Zentralblatt MATH: 1044.60092
    Digital Object Identifier: doi:10.1214/aop/1019160117
  • [7] Cox, J. T. and Perkins, E. A. (2014). A complete convergence theorem for voter model perturbations. Ann. Appl. Probab. 24 150–197.
    Zentralblatt MATH: 1291.60202
    Digital Object Identifier: doi:10.1214/13-AAP919
  • [8] Erdős, P. and Ney, P. (1974). Some problems on random intervals and annihilating particles. Ann. Probab. 2 828–839.
  • [9] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Springer, Berlin.
    Zentralblatt MATH: 0412.60095
  • [10] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [11] Halperin, B. I. and Weinrib, A. (1983). Critical phenomena in systems with long-range-correlated quenched disorder. Phys. Rev. B 27 413–427.
  • [12] Hara, T., van der Hofstad, R. and Slade, G. (2003). Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 349–408.
    Zentralblatt MATH: 1044.82006
    Digital Object Identifier: doi:10.1214/aop/1046294314
  • [13] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [14] Holley, R. A. and Liggett, T. M. (1975). Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3 643–663.
    Zentralblatt MATH: 0367.60115
    Digital Object Identifier: doi:10.1214/aop/1176996306
  • [15] Holmes, M., Mohylevskyy, Y. and Newman, C. M. (2015). The voter model chordal interface in two dimensions. J. Stat. Phys. 159 937–957.
    Mathematical Reviews (MathSciNet): MR3336987
    Zentralblatt MATH: 1328.82024
    Digital Object Identifier: doi:10.1007/s10955-015-1198-9
  • [16] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
    Zentralblatt MATH: 0892.60001
  • [17] Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston, MA.
    Zentralblatt MATH: 0522.60097
  • [18] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd ed. Imperial College Press, London.
    Zentralblatt MATH: 1077.60001
  • [19] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
    Zentralblatt MATH: 1228.60004
  • [20] Lebowitz, J. L. and Saleur, H. (1986). Percolation in strongly correlated systems. Phys. A 138 194–205.
    Zentralblatt MATH: 0666.60110
    Digital Object Identifier: doi:10.1016/0378-4371(86)90180-9
  • [21] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [22] Liggett, T. M. and Steif, J. E. (2006). Stochastic domination: The contact process, Ising models and FKG measures. Ann. Inst. Henri Poincaré Probab. Stat. 42 223–243.
    Zentralblatt MATH: 1087.60074
    Digital Object Identifier: doi:10.1016/j.anihpb.2005.04.002
  • [23] Marinov, V. (2007). Percolation in correlated systems. Ph.D. thesis, Rutgers University.
  • [24] Marinov, V. I. and Lebowitz, J. L. (2006). Percolation in the harmonic crystal and voter model in three dimensions. Phys. Rev. E (3) 74 031120, 7.
  • [25] Peierls, R. (1936). On Ising’s model of ferromagnetism. Math. Proc. Cambridge Philos. Soc. 32 477–481.
    Zentralblatt MATH: 0014.33604
    Digital Object Identifier: doi:10.1017/S0305004100019174
  • [26] Perkins, E. (2002). Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
    Zentralblatt MATH: 1020.60075
  • [27] Perkins, E. A. (1995). Measure-valued branching diffusions and interactions. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1036–1046. Birkhäuser, Basel.
    Zentralblatt MATH: 0844.60057
  • [28] Ráth, B. (2015). A short proof of the phase transition for the vacant set of random interlacements. Electron. Commun. Probab. 20 no. 3, 11.
  • [29] Rodriguez, P.-F. and Sznitman, A.-S. (2013). Phase transition and level-set percolation for the Gaussian free field. Comm. Math. Phys. 320 571–601.
    Zentralblatt MATH: 1269.82028
    Digital Object Identifier: doi:10.1007/s00220-012-1649-y
  • [30] Schwartz, D. (1978). On hitting probabilities for an annihilating particle model. Ann. Probab. 6 398–403.
    Zentralblatt MATH: 0404.60098
    Digital Object Identifier: doi:10.1214/aop/1176995526
  • [31] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
    Zentralblatt MATH: 1168.60036
    Digital Object Identifier: doi:10.1002/cpa.20267
  • [32] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
    Zentralblatt MATH: 1202.60160
  • [33] Sznitman, A.-S. (2012). Decoupling inequalities and interlacement percolation on $G\times\mathbb{Z}$. Invent. Math. 187 645–706.
    Zentralblatt MATH: 1277.60183
    Digital Object Identifier: doi:10.1007/s00222-011-0340-9
  • [34] van den Berg, J. (2011). Sharpness of the percolation transition in the two-dimensional contact process. Ann. Appl. Probab. 21 374–395.
    Zentralblatt MATH: 1247.60136
    Digital Object Identifier: doi:10.1214/10-AAP702
  • [35] van den Berg, J., Björnberg, J. E. and Heydenreich, M. (2015). Sharpness versus robustness of the percolation transition in 2d contact processes. Stochastic Process. Appl. 125 513–537.
  • [36] Weinrib, A. (1984). Long-range correlated percolation. Phys. Rev. B (3) 29 387–395.
  • [37] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press, Cambridge.

The Institute of Mathematical Statistics

The Annals of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more