The Cook Book
Recipe for the week of December 12 - 18
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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