The Annals of Probability

The Threshold Voter Automaton at a Critical Point

Jeffrey E. Steif

Full-text: Open access


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.

Article information

Ann. Probab. Volume 22, Number 3 (1994), 1121-1139.

First available in Project Euclid: 19 April 2007

Permanent link to this document


Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations

Cellular automata critical value Chen-Stein method branching processes


Steif, Jeffrey E. The Threshold Voter Automaton at a Critical Point. Ann. Probab. 22 (1994), no. 3, 1121--1139. doi:10.1214/aop/1176988597.

Export citation