## The Annals of Probability

### Percolation on the stationary distributions of the voter model

#### 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
Revised: February 2016
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835234

Digital Object Identifier
doi:10.1214/16-AOP1104

Mathematical Reviews number (MathSciNet)
MR3650418

Zentralblatt MATH identifier
06754789

#### 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

#### References

• [1] Arratia, R. (1983). Site recurrence for annihilating random walks on $\textbf{Z}_{d}$. Ann. Probab. 11 706–713.
• [2] Bramson, M., Cox, J. T. and Le Gall, J.-F. (2001). Super-Brownian limits of voter model clusters. Ann. Probab. 29 1001–1032.
• [3] Bramson, M. and Griffeath, D. (1980). Asymptotics for interacting particle systems on $\textbf{Z}^{d}$. Ann. Probab. 7 418–432.
• [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.
• [5] Clifford, P. and Sudbury, A. (1973). A model for spatial conflict. Biometrika 60 581–588.
• [6] Cox, J. T., Durrett, R. and Perkins, E. A. (2000). Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28 185–234.
• [7] Cox, J. T. and Perkins, E. A. (2014). A complete convergence theorem for voter model perturbations. Ann. Appl. Probab. 24 150–197.
• [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.
• [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.
• [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.
• [15] Holmes, M., Mohylevskyy, Y. and Newman, C. M. (2015). The voter model chordal interface in two dimensions. J. Stat. Phys. 159 937–957.
• [16] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
• [17] Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston, MA.
• [18] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd ed. Imperial College Press, London.
• [19] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
• [20] Lebowitz, J. L. and Saleur, H. (1986). Percolation in strongly correlated systems. Phys. A 138 194–205.
• [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.
• [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.
• [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.
• [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.
• [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.
• [30] Schwartz, D. (1978). On hitting probabilities for an annihilating particle model. Ann. Probab. 6 398–403.
• [31] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
• [32] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
• [33] Sznitman, A.-S. (2012). Decoupling inequalities and interlacement percolation on $G\times\mathbb{Z}$. Invent. Math. 187 645–706.
• [34] van den Berg, J. (2011). Sharpness of the percolation transition in the two-dimensional contact process. Ann. Appl. Probab. 21 374–395.
• [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.