We investigate majority rule dynamics in a population with two classes of people, each with two opinion states $\pm 1$, and with tunable interactions between people in different classes. In an update, a randomly selected group adopts the majority opinion if all group members belong to the same class; if not, majority rule is applied with rate $\epsilon$. Consensus is achieved in a time that scales logarithmically with population size if $\epsilon \ge {\epsilon }_c=\frac{1}{9}$. For $\epsilon <{\epsilon }_c$, the population can get trapped in a polarized state, with one class preferring the $+1$ state and the other preferring $-1$. The time to escape this polarized state and reach consensus scales exponentially with population size.

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• Nonequilibrium statistical mechanics
• Bifurcation analysis
• Extreme event statistics
• Stochastic analysis methods