Majority versus minority dynamics: Phase transition in an interacting two-state spin system

We introduce a simple model of opinion dynamics in which binary-state agents evolve due to the influence of agents in a local neighborhood. In a single update step, a fixed-size group is defined and all agents in the group adopt the state of the local majority with probability p or that of the local minority with probability $1-p.\mathrm{}$ For group size $G=3,$ there is a phase transition at $p_c=2/3\mathrm{}$ in all spatial dimensions. For $p>\mathrm{}{p}_c,$ the global majority quickly predominates, while for $p<\mathrm{}{p}_c,$ the system is driven to a mixed state in which the densities of agents in each state are equal. For ${p=p}_c,$ the average magnetization (the difference in the density of agents in the two states) is conserved and the system obeys classical voter model dynamics. In one dimension and within a Kirkwood decoupling scheme, the final magnetization in a finite-length system has a nontrivial dependence on the initial magnetization for all $p\ne p_c,$ in agreement with numerical results. At $p_c,$ the exact two-spin correlation functions decay algebraically toward the value 1 and the system coarsens as in the classical voter model.