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Phase Separation in a Nonlinear Voter Model

We learned this two-parameter family of voting rules (1 = red = Republican, 0 = blue = Democrats) from Jane Molofsky, Rick Durrett and Si Levin. The neighbor set is 'von Neumann' (N,S,E,W, and Center). p(k) is the probability that the voter at x takes opinion 1 next time when k voters in the neighborhood (x included) have opinion 1 this time. By assumption, p(0)=0 and p(5-k)=1-p(k), so that the rule is symmetric with 'all 1's' and 'all 0's' as traps.

This soup shows phase separation in the Nonlinear Voter Model with parameters p(1)=.31, p(2)=0, at time 2000 on a 500x400 array. The system, which started from uniform symmetric 2-color noise, is clearly clustering. But each cluster (predominantly Republican, predominantly Democrat) is sprinkled with a minority representation of the opposite opinion. Similar behavior arises in certain stochastic Ising models and perturbed majority vote systems, but this is the first 2-state nearest neighbor example we know of in which the total consensus states are absorbing.

Non-trivial phase separation occurs in a remarkably small region of the (p(1),p(2)) NLVM phase space -- less that 1/4 of 1% of the unit square. We have made a careful empirical estimate of the phase boundaries based on extensive CAM8 experimentation. The resulting phase portrait is shown in vmphases.gif. Note the miniscule 'Bermuda triangle' of coexistence.

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