Coexistence or Consensus?

In the voter model each site has a type from a set S of possibilities. A reasonable choice for the two party system in the United States is S = {0,1}, but one can adapt to other countries by using larger finite sets. In developing the theory it is convenient to consider the model starting with state_0[x] = x, i.e., initially all sites are different. Finally, for some applications it is useful to choose S to be the unit interval (0,1), so that we can pick new types at random and with probability one not duplicate an existing type.

The dynamics are perhaps the simplest possible: each site x at times of a rate 1 Poisson process wakes up, picks one of the four nearest neighbors at random and imitates its opinion, i.e., we set state_t[x] = state_t[y].

The voter model was formulated by Kimura and Weiss (1964) and studied by geneticists (e.g., Rohlf and Schnell (1971)) long before it was rediscovered by mathematicians. However, the Basic Dichotomy is due to Holley and Liggett (1975).

Theorem. (i) In dimensions 1 and 2 the system approaches complete consensus. That is, for any set of types S and any initial condition, the probability state_t[x] = state_t[y] tends to 1.

(ii) In dimensions > 2, differences of opinion can persist. For example if S = { 0, 1 } and we start with product measure with density p (i.e., the states of different sites are independent and equal to 1 with probability p) then the limit as t tends to infinity is a translation invariant stationary distribution mu_p. Furthermore, all stationary distributions are convex combinations of the mu_p.

Sketch of Proof. We begin by giving a special construction of the process. Introduce independent Poisson processes for each site x and at these times draw an arrow from x to a randomly chosen neighbor y to indicate that x adopts the opinion at y. The reason for using this construction is that it allows us to work backwards in time. It follows from the definition of the model that the state at x at time t is the same at that of exactly one site W(x,t;s) at time t-s. This site will stay the same until the tail of an arrow hits its location. At this point the site in question imitated the neighbor at the head so W(x,t;s) jumps there. For an example see the figure below where the paths W(x,t;s) and W(y,t;s) are drawn with broad gray lines:

W(x,t;s) is a continuous time random walk, i.e., it stays at a site for an exponential amount of time with mean 1, then jumps to a randomly chosen neighbor. If we consider two sites x and y then the difference in position between their ancestral lines, W(x,t;s) - W(y,t;s), is again a continuous time random walk, which now jumps at rate 2. In dimensions 1 and 2 the difference random walk will hit 0 with probability 1 at some time and then the two ancestral lines will agree at later times. When W(x,t;s) and W(y,t;s) hit by time t (as in the picture above) the two sites will agree at time t. The first result follows.

Using the fact that two random walks have positive probabiity of never hitting in dimensions d > 2, one comes easily to the existence of the stationary distributions in (ii). The uniqueness result is considerably more subtle. See the original paper of Holley and Liggett (1975) or Chapter V of Liggett (1985).

Nonspatial Models. One can of course consider the voter model with a finite set of individuals that have no spatial structure. In this case working backwards in time leads to the coalescent which has been extensively studied. See Kingman (1980), Hudson (1990), Donnelly and Tavare (1995) and references therein.

Kimura, M. and Weiss, G.H. (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 313-326

Rohlf, F.J. and Schnell, G.D. (1971) An investigation of the isolation by distance model. Am. Nat. 105, 295-324

Holley, R.A. and Liggett, T.M. (1975) Ergodic theorems for weakly interacting systems and the voter model. Ann. Prob. 3, 643-663

Kingman, J.F.C. (1980) Mathematics of Genetic Diversity. SIAM Lecture Notes #34.

Liggett, T.M. (1985) Interacting Particle Systems. Springer, New York

Hudson, R.R. (1991) Gene genealogies and the coalescent process. Pages 1-44 in Volume 7 of Oxford Surveys in Evolutionary Biology, edited by D. Futuyama and J. Antonovics.

Donnelly, P., and Tavare, S. (1995) Coalescents and genealogical structure under neutrality. Ann. Rev. Genetics. 29, 401-421

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