Consensus and diversity in multistate noisy voter models

We study a variant of the voter model with multiple opinions; individuals can imitate each other and also change their opinion randomly in mutation events. We focus on the case of a population with all-to-all interaction. A noise-driven transition between regimes with multimodal and unimodal stationary distributions is observed. In the former, the population is mostly in consensus states; in the latter, opinions are mixed. We derive an effective death-birth process, describing the dynamics from the perspective of one of the opinions and use it to analytically compute marginals of the stationary distribution. These calculations are exact for models with homogeneous imitation and mutation rates and an approximation if rates are heterogeneous. Our approach can be used to characterize the noise-driven transition and to obtain mean switching times between consensus states.