Nonlinear Voter Model

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 x will be occupied with probability p[k(x)] at time t+1.

(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

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