The nearest neighbor voter model is very easy to understand in one dimension. If we start with all sites different then at any time the set of sites with a given opinion will be an interval [a(t),b(t)] or the emptyset. It is easy to see that while the interval is nonempty the endpoints will move left at rate 1/2 and right at rate 1/2.
Since the endpoints are random walks it is easy to guess that most intervals will have length of order square root of t, and hence the voter model on a ring of L sites will take about L^2 (L squared) steps to reach total consensus. These guesses are not difficult to prove. See Bramson and Griffeath (1980) or Section V.3 of Liggett (1985) for precise statements.
s3 Exercise. Start the one dimensional voter model with eight colors (our maximum) and range 1. Quickly the system will be reduced to several intervals that perform random walks.
Suppose now that the range r is bigger than 1, i.e., two sites x and y are neighbors if the distance between them is less than r. If we start with opinion 0 at all sites x > 0 and opinion 1 at all other sites then at any time t there is always a leftmost 0, l_t, and a right most 1, r_t. The "confused" region, (l_t,r_t), which is always the empty set in the nearest neighbor case, could conceivably grow large with time, but Cox and Durrett (1995) have shown
Theorem. P( r_t - l_t = k) converges to a limit probability distribution p_k.
s3 Exercise. Set the range to be some 5 and pick the rectangle initial configuration with width = 1/2 the size of your system to demonstrate this result. It is known that the limiting interface width distribution p_k has infinite mean. Do the simulations confirm or deny this?
Bramson, M. and Griffeath, D. (1980) Clustering and dispersion rates for some interacting particle systems on Z. Ann. Prob. 8, 183-213
Liggett, T.M. (1985) Interacting Particle Systems. Springer, New York
Cox, J.T. and Durrett, R. (1995) Hybrid zones and voter model interfaces. Bernoulli 1, 343-370
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