Limit behavior of the exponential voter model

The recent research on opinion dynamics within communities has concentrated on the so-called voter models [1, 3, 6]. Formally, these are stochastic dynamical systems working as follows. One is given a lattice or graph whose vertices are called cells or sites. Each site x has a set of neighbor sites including the cell itself that constitute the neighborhood of x and which may influence the state of the original site, which is initially either On or Off. Uniformly, a site x is chosen at random and its state is updated according to the following rule. One counts the number of On sites in the neighborhood of the chosen site call this k(x). The site then turns (or stays) On with probability f(k(x)), where f is predefined. One then repeats this process by picking another site, updating, and so on. This generates a sequence of configurations whose long-term behavior is of certain interest. The function f is usually chosen to be increasing so that a site is more and more likely to be On if more and more of its neighboring sites are On. That would correspond to a majority-biased behavior. According to the modeling, the increase of f may be either linear or non-linear (exponential, in particular). Also, the function is chosen so that the system is symmetric under color change, meaning that if we invert the state of each cell, apply the update process, and invert the system again, we should get the same result (probabilistically) as if we simply updated the system.