The Threshold Voter Automaton at a Critical Point

Abstract

We consider the threshold voter automaton in one dimension with threshold $\tau > n/2$, where $n$ is the number of neighbors and where we start from a product measure with density $\frac{1}{2}$. It has recently been shown that there is a critical value $\theta_c \approx 0.6469076$, so that if $\tau = \theta n$ with $\theta > \theta_c$ and $n$ is large, then most sites never flip, while for $\theta \in (\frac{1}{2}, \theta_c)$ and $n$ large, there is a limiting state consisting mostly of large regions of points of the same type. Using a supercritical branching process, we show that the behavior at $\theta_c$ differs from both the $\theta > \theta_c$ regime and the $\theta < \theta_c$ regime and that, in some sense, there is a discontinuity both from the left and from the right at this critical value.