## One Dimensional Voter Model

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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