Percolation on the stationary distributions of the voter model with stirring

Abstract

The voter model with stirring is a variant of the classical voter model on ${\mathbb{Z}}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf{v}\ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d\ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{{\mu }_{\alpha ,\mathsf{v}}:\alpha \in [0,1]\}$, where~$\alpha$ is the density of voters with opinion~1. Sampling a configuration~$\xi$ from~${\mu }_{\alpha ,\mathsf{v}}$, we study~$\xi$ as a site percolation model on~${\mathbb{Z}}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$\xi$. Letting~${\alpha }_c(\mathsf{v})$ be the supremum of all the values of~$\alpha$ for which percolation does not occur~${\mu }_{\alpha ,\mathsf{v}}$-a.s., we prove that ${\alpha }_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf{v}$ large enough, the model exhibits a non-trivial phase transition in~$\alpha$.

32 pages, 1 figure