The
voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the
contact process. In this process \eta(x) is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform.
Discrete time process In the discrete time voter model in one dimension, \xi_t(x): \mathbb{Z} \to \{0,1\} represents the state of particle x at time t. Informally each individual is arranged on a line and can "see" other individuals that are within a radius, r. If more than a certain proportion, \theta of these people disagree then the individual changes her attitude, otherwise she keeps it the same.
Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value \theta_c such that if \theta > \theta_c most individuals never change, and for \theta \in (1/2, \theta_c) in the limit most sites agree. (Both of these results assume the probability of \xi_0(x) = 1 is one half.) This process has a natural generalization to more dimensions, some results for this are discussed in
Durrett and Steif (1993).
Continuous time process The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by
Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and
Liggett (1975). This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time. == References ==