Percolation on the stationary distributions of the voter model with stirring

TL;DR

This paper investigates the percolation properties of stationary distributions in a stirring voter model on  lattice, showing convergence to classical percolation thresholds as stirring increases.

Contribution

It establishes the convergence of the critical density for percolation in the voter model to the classical Bernoulli threshold as stirring rate grows.

Findings

 critical density (\u211d) is approached as (\u211d, v) converges to () when v   infinity.

For large v, the model exhibits a non-trivial phase transition in opinion density .

The set of extremal stationary measures is characterized for d  3 and any v.

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~$\alpha$.