Genre: Information/Explanation

voter model coexists on the graph with two clusters One can imagine that there is a " voter " at each point on a connected graph , where the connections indicate that there is some form of interaction between a pair of voters ( nodes ) . The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours . A voter's opinion at any given time can take one of two values , labelled 0 and 1 . At random times , a random individual is selected and that voter's opinion are changed according to a stochastic rule . Specifically , for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual's opinion is transferred to the chosen voter . An alternative interpretation is in terms of spatial conflict . Suppose two nations control the areas ( sets of nodes ) labelled 0 or 1 . A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation . Note that only one flip happens each time . Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains . Frequently , these problems will then be reduced to others involving independent Markov chains . What we are interested in is the limiting behavior of the models . Since the flip rates of a site depends its neighbours , it is obvious that when all sites take the same value , the whole system stops changing forever . Therefore , a voter model has two trivial extremal stationary distributions , the point-masses δ 0 and δ 1 on η ≡ 0 or η ≡ 1 respectively , which represent consensus . The main question we will discuss is whether or not there are others , which would then represent coexistence of different opinions in equilibrium . We say that coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0 's and 1 's .