Fixation Results for Threshold Voter Systems

Abstract

We consider threshold voter systems in which the threshold τ>n/2, where n is the number of neighbors, and we present results in support of the following picture of what happens starting from product measure with density 1/2. The system fixates, that is, each site flips only finitely many times. There is a critical value, θc, so that if τ=θn with θ>θc and n is large then most sites never flip, while for θ∈(1/2,θc) and n large, the limiting state consists mostly of large regions of points of the same type. In d=1,θc≈0.6469076 while in d>1,θc=3/4.