## The Annals of Probability

### Fixation Results for Threshold Voter Systems

#### Abstract

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 $d > 1, \theta_c = 3/4$.

#### Article information

Source
Ann. Probab. Volume 21, Number 1 (1993), 232-247.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176989403

Digital Object Identifier
doi:10.1214/aop/1176989403

Mathematical Reviews number (MathSciNet)
MR1207225

Zentralblatt MATH identifier
0769.60092

JSTOR

#### Citation

Durrett, Richard; Steif, Jeffrey E. Fixation Results for Threshold Voter Systems. Ann. Probab. 21 (1993), no. 1, 232--247. doi:10.1214/aop/1176989403. http://projecteuclid.org/euclid.aop/1176989403.