# Analysis, Geometry, and Dynamical Systems Seminar

### Percolation on the stationary distribution of the voter model on $\mathbb{Z}^d$

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 the model is considered in dimension 3 or higher, its set of (extremal) stationary distributions is equal to a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a 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 from the point of view of site percolation on ${\mathbb Z}^d$. We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in $\alpha$. If the voter model is allowed to have long range, we prove the same result for dimensions 3 and higher. Joint work with Balázs Ráth.