The Cook Book
Recipe for the week of December 12  18
Phase Separation in a Nonlinear Voter Model
 We learned this twoparameter 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(5k)=1p(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 2color 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 2state nearest neighbor example we know of in which the total consensus states are absorbing.
 Nontrivial 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.
