and implies if . Clustering and coexistence The interest in is the limiting behavior of the models. Since the flip rates of a site depends on 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 and on or respectively, which represent consensus. The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium. It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's. On the other hand, if for all and all initial configurations, then It is said that clustering occurs. It is important to distinguish clustering with the concept of cluster. Clusters are defined to be the connected components of or . The linear voter model Model description This section will be dedicated to one of the basic voter models, the Linear Voter Model. If •,• be the transition probabilities for an irreducible random walk on , then: Then in Linear voter model, the transition rates are linear functions of : Or if indicates that a flip happens at , then transition rates are simply: A process of coalescing random walks is defined as follows. Here denotes the set of sites occupied by these random walks at time . To define , consider several (continuous time) random walks on with unit exponential holding times and transition probabilities •,•, and take them to be independent until two of them meet. At that time, the two that meet coalesce into one particle, which continues to move like a random walk with transition probabilities •,• . The concept of Duality is essential for analysing the behavior of the voter models. The linear voter models satisfy a very useful form of duality, known as coalescing duality, which is: where is the initial configuration of and is the initial state of the coalescing random walks . Limiting behaviors of linear voter models Let be the transition probabilities for an irreducible random walk on and , then the duality relation for such linear voter models says that where and are (continuous time) random walks on with , , and is the position taken by the random walk at time . and forms a coalescing random walks described at the end