below: The theorem follows when letting . The threshold voter model Model description This section, concentrates on a kind of non-linear voter models, known as the threshold voter model. To define it, let be a neighbourhood of that is obtained by intersecting with any compact, convex, symmetric set in ; in other word, is assumed to be a finite set that is symmetric with respect to all reflections and irreducible (i.e. the group it generates is ). It can always be assumed that contains all the unit vectors . For a positive integer , the threshold voter model with neighbourhood and threshold is the one with rate function: Simply put, the transition rate of site is 1 if the number of sites that do not take the same value is larger or equal to the threshold T. Otherwise, site stays at the current status and will not flip. For example, if , and , then the configuration is an absorbing state or a trap for the process. Limiting behaviors of threshold voter model If a threshold voter model does not fixate, the process should be expected to will coexist for small threshold and cluster for large threshold, where large and small are interpreted as being relative to the size of the neighbourhood, . The intuition is that having a small threshold makes it easy for flips to occur, so it is likely that there will be a lot of both 0's and 1's around at all times. The following are three major results: If , then the process fixates in the sense that each site flips only finitely often. If and , then the process clusters. If with sufficiently small() and sufficiently large, then the process coexists. Here are two theorems corresponding to properties (1) and (2). Theorem 3.1 If , then the process fixates. Theorem 3.2 The threshold voter model in one dimension () with , clusters. proof The idea of the proof is to construct two sequences of random times , for with the following properties: , are i.i.d.with , are i.i.d.with , the random variables in (b) and (c) are independent of each other, event A= is constant on , and event A holds for every . Once this construction is made, it will follow from renewal theory that Hence,, so that the process clusters. Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if