## The Annals of Probability

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

### Fixation Results for Threshold Voter Systems

Richard Durrett and Jeffrey E. Steif

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Cellular automata large deviations voter models

#### 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.