Constrained opinion dynamics: freezing and slow evolution

We study opinion formation in a population that consists of leftists, centrists, and rightist. In an interaction between neighboring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist). In contrast, leftists and rightists do not affect each other. The initial density of centrists rho₀ controls the evolution. With probability rho₀ the system reaches a centrist consensus, while with probability 1-rho₀ a frozen population of leftists and rightists results. In one dimension, we determine this frozen state and the opinion dynamics by mapping the system onto a spin-1 Ising model with zero-temperature Glauber kinetics. In the frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x^-2 (1-psi) and the average domain size grows as L^2*psi, where L is the system length. The approach to this frozen state is governed by a t^-psi long-time tail with psi-->2*rho₀/pi as rho₀-->0.