## Electronic Communications in Probability

### On the threshold of spread-out voter model percolation

#### Abstract

In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z} ^d$ has state (or ‘opinion’) 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu _{\alpha , R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0’s and 1’s in which the density of 1’s is $\alpha$ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z} ^d$, focusing on asymptotics as $R \to \infty$. In [RV15], we have shown that, if $R$ is large, there is a critical value $\alpha _c(R)$ such that there is percolation if $\alpha > \alpha _c(R)$ and no percolation if $\alpha < \alpha _c(R)$. Here we prove that, as $R \to \infty$, $\alpha _c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z} ^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu _{\alpha ,R}$ which is of independent interest.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 50, 12 pp.

Dates
Accepted: 14 August 2017
First available in Project Euclid: 6 October 2017

https://projecteuclid.org/euclid.ecp/1507255233

Digital Object Identifier
doi:10.1214/17-ECP80

Mathematical Reviews number (MathSciNet)
MR3710806

Zentralblatt MATH identifier
06797803

#### Citation

Ráth, Balázs; Valesin, Daniel. On the threshold of spread-out voter model percolation. Electron. Commun. Probab. 22 (2017), paper no. 50, 12 pp. doi:10.1214/17-ECP80. https://projecteuclid.org/euclid.ecp/1507255233

#### References

• [BGP] Benjamini, I., Gurel-Gurevich, O., and Peled, R. On K-wise Independent Distributions and Boolean Functions. arXiv:1201.3261
• [BLM87] Bricmont, J., Lebowitz, J. and Maes, C. Percolation in strongly correlated systems: the massless Gaussian field. Journal of statistical physics 48, no. 5 (1987): 1249-1268.
• [CS73] Clifford, P., and Sudbury, A. A model for spatial conflict. Biometrika 60, no. 3 (1973): 581-588.
• [DP96] Deuschel, J-D., and Pisztora, A. Surface order large deviations for high-density percolation. Probability Theory and Related Fields 104, no. 4 (1996): 467-482.
• [Gr99] Grimmett, G. Percolation. Springer-Verlag Berlin (Second edition) (1999).
• [HL75] Holley, R., and Liggett, T. Ergodic theorems for weakly interacting infinite systems and the voter model. The Annals of probability (1975): 643-663.
• [LS88] Lebowitz, J. L., and Schonmann, R. H. Pseudo-free energies and Large deviations for Non Gibbsian FKG measures. Probability Theory and Related Fields 77.1 (1988): 49-64.
• [LS86] Lebowitz, J., and Saleur, H. Percolation in strongly correlated systems. Physica A: Statistical Mechanics and its Applications 138, no. 1-2 (1986): 194-205.
• [Li85] Liggett, T. Interacting particle systems. Grundlehren der mathematischen Wissenschaften 276, Springer (1985).
• [Ma07] Marinov, V. Percolation in correlated systems. PhD Thesis, Rutgers The State University of New Jersey-New Brunswick, 2007.
• [ML06] Marinov, V., and Lebowitz, J. Percolation in the harmonic crystal and voter model in three dimensions. Physical Review E 74, no. 3 (2006): 031120.
• [Pi96] Pisztora, A. Surface order large deviations for Ising, Potts and percolation models. Probability Theory and Related Fields 104, no. 4 (1996): 427-466.
• [PR15] Popov, S., and Ráth, B. On decoupling inequalities and percolation of excursion sets of the Gaussian free field. J. of Stat. Phys., (2015), 159 (2), 312-320.
• [PT15] Popov, S., and Teixeira, A. Soft local times and decoupling of random interlacements. J. European Math. Soc. 17 (10), 2545-2593 (2015).
• [Ra15] Ráth, B. A short proof of the phase transition for the vacant set of random interlacements. Electronic Communications in Probability 20 (2015).
• [RV15] Ráth, B., and Valesin, D. Percolation on the stationary distributions of the voter model. Annals of Probability 45 (3), 1899-1951 (2017).
• [RS13] Rodriguez, P.-F., and Sznitman, A.-S. Phase transition and level set percolation for the Gaussian free field. Communications in Mathematics Physics 320 (2), 571-601 (2013).
• [R17] Rodriguez, P.-F., Decoupling inequalities for the Ginzburg-Landau $\nabla \varphi$ models. arXiv:1612.02385
• [ST17] Steif, J., Tykesson, J. Generalized Divide and Color models. arXiv:1702.04296
• [Sz12] Sznitman, A.-S. Decoupling inequalities and interlacement percolation on $G \times \mathbb Z$. Inventiones mathematicae, 187, 3, 645-706 (2012).