Fixation Results for Threshold Voter Systems

We consider threshold voter systems in which the threshold $\tau > n/2$, where $n$ is the number of neighbors, and we present results in support of the following picture of what happens starting from product measure with density $1/2$. The system fixates, that is, each site flips only finitely many times. There is a critical value, $\theta_c$, so that if $\tau = \theta n$ with $\theta > \theta_c$ and $n$ is large then most sites never flip, while for $\theta \in (1/2, \theta_c)$ and $n$ large, the limiting state consists mostly of large regions of points of the same type. In $d = 1, \theta_c \approx 0.6469076$ while in $d > 1, \theta_c = 3/4$.