Percolation on the stationary distributions of the voter model.(English)Zbl 1437.60076

Summary: 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 $$d\geq3$$.

MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 82B43 Percolation
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