of section 2.1. is a symmetrized random walk. If is recurrent and , and will hit eventually with probability 1, and hence Therefore, the process clusters. On the other hand, when , the system coexists. It is because for , is transient, thus there is a positive probability that the random walks never hit, and hence for for some constant corresponding to the initial distribution. If be a symmetrized random walk, then there are the following theorems: Theorem 2.1 The linear voter model clusters if is recurrent, and coexists if is transient. In particular, the process clusters if and , or if and ; the process coexists if . Remarks: To contrast this with the behavior of the threshold voter models that will be discussed in next section, note that whether the linear voter model clusters or coexists depends almost exclusively on the dimension of the set of sites, rather than on the size of the range of interaction. Theorem 2.2 Suppose is any translation spatially ergodic and invariant probability measure on the state space , then If is recurrent, then ; If is transient, then . where is the distribution of ; means weak convergence, is a nontrivial extremal invariant measure and . A special linear voter model One of the interesting special cases of the linear voter model, known as the basic linear voter model, is that for state space : So that In this case, the process clusters if , while coexists if . This dichotomy is closely related to the fact that simple random walk on is recurrent if and transient if . Clusters in one dimension d = 1 For the special case with , and for each . From Theorem 2.2, , thus clustering occurs in this case. The aim of this section is to give a more precise description of this clustering. As mentioned before, clusters of an are defined to be the connected components of or . The mean cluster size for is defined to be: provided the limit exists. Proposition 2.3 Suppose the voter model is with initial distribution and is a translation invariant probability measure, then Occupation time Define the occupation time functionals of the basic linear voter model as: Theorem 2.4 Assume that for all site x and time t, , then as , almost surely if proof By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation