This state of each site at time t can be either 0 or 1, which should be thought of as indicating the type of the particle there. The name comes from thinking 0=Democrat, 1=Republican, but one can also think of two competing species.
In this model time is discrete t = 0, 1, 2, ... and we move from time t to t+1 by the following procedure:
(i) Add up the states of x and its four nearest neighbors,
or in other words count the number of 1's. If the result is
(ii) Once we have the probabilities p[k(x)], the choices
of state for the various sites are made independently.
As written above this model has six parameters. However, we will assume
(a) p[k] = p[5-k] so that the system is symmetric under the interchange
of 0's and 1's.
(b) p[0]=0 so that all 0's is an absorbing
state, i.e., once we enter it we cannot leave. Of course, by symmetry
p[5]=1 and all 1's is also an absorbing state.
(a) and (b) leave us with two parameters, p[1] and p[2], and bring
us to our main question:
For what parameter values is coexistence possible?
To be precise we want to know if there is a stationary
distribution that concentrates on configurations
with infinitely many 1's and infinitely many 0's.
One way to try to construct one is to examine the limiting behavior
starting from product measure
with density 1/2, i.e., from the initial configuration in which
sites are independent, and equal to 0 or 1 with probability 1/2 each.
Our plan will be foiled if clustering occurs, i.e., if for
each pair of sites x and y we have
P( state_t[x] = 1, state_t[y] = 0 ) approaches 0 as t
tends to infinity
Empirically the boundary between coexistence and clustering is given
by the Levin line: 3 p[1] + p[2] = 1.
s3 Exercises.Some interesting parameter values to investigate are:
(a) Threshold Voter Model. p[1] = p[2] = 0.5:
as long as there is at least one 0 and at least one
1 in the neighborhood of a site it flips a coin.
For results about this model in continuous time, see
Liggett (1994).
(b) Majority Vote Model. p[1] = p[2] = 0. Site flips to the
majority in its neighborhood. This process gets stuck in an
absorbing state, so it more interesting to look at what happens
when there is a small amount of noise, i.e., p[0] = p[1] = p[2] = 0.05.
Note that we have ignored assumption (b) p[0] = 0.
(c) Colorado Springs Voter Model. p[1] = 0.3, p[2] = 0. This process
is named after the location of the conference where the system
was invented and simulated on David Griffeath's laptop (using
his software WinCA). Here there is clustering with growing regions
that are mostly 1's or mostly 0's but in each maintains a positive
fraction of the minority type. It would be very interesting to
show that a small perturbation of this model will all 0 < p[i]
< 1 has two translation invariant stationary distribution.
Related Work. For nonlinear voter models
in continuous time see Cox and Durrett (1993).
Cox, J.T. and Durrett, R. (1991) Nonlinear voter models. Pages 189-202
in Random Walks, Brownian Motion, and Interacting Particle Systems.
Edited by R. Durrett and H. Kesten. Birkhauser, Boston.
Liggett, T.M. (1994) Coexistence in threshold voter models. Ann.
Prob. 22, 764-802