. For example, take and . If is constant on alternating vertical infinite strips, that is for all : then no transition ever occur, and the process fixates. (b) Under the assumption of Theorem 3.2, the process does not fixate. To see this, consider the initial configuration , in which infinitely many zeros are followed by infinitely many ones. Then only the zero and one at the boundary can flip, so that the configuration will always look the same except that the boundary will move like a simple symmetric random walk. The fact that this random walk is recurrent implies that every site flips infinitely often. Property 3 indicates that the threshold voter model is quite different from the linear voter model, in that coexistence occurs even in one dimension, provided that the neighbourhood is not too small. The threshold model has a drift toward the "local minority", which is not present in the linear case. Most proofs of coexistence for threshold voter models are based on comparisons with hybrid model known as the threshold contact process with parameter . This is the process on with flip rates: Proposition 3.3 For any and , if the threshold contact process with has a nontrivial invariant measure, then the threshold voter model coexists. Model with threshold T = 1 The case that is of particular interest because it is the only case in which it is known exactly which models coexist and which models cluster. In particular, there is interest in a kind of Threshold T=1 model with that is given by: can be interpreted as the radius of the neighbourhood ; determines the size of the neighbourhood (i.e., if , then ; while for , the corresponding ). By Theorem 3.2, the model with and clusters. The following theorem indicates that for all other choices of and , the model coexists. Theorem 3.4 Suppose that , but . Then the threshold model on with parameter coexists. The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett. See also Probabilistic Cellular Automata sequential dynamical system contact process Notes References Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999. Stochastic models Lattice models