Barrier induced stalemate-consensus transition of self-propelled participants subject to majority rule

Natural or artificial barriers, such as the Himalayas, the Berlin Wall, or the Korean Demilitarized Zone, can significantly impede human migration. As a consequence, they may also hinder the dissemination of opinions within society, thereby contributing to divergent geopolitical landscapes and cultural developments. This raises a fundamental question: how do such barriers influence the opinion dynamics of mobile agents, such as human beings? In particular, can a barrier induce transitions in collective opinion states among spatially segregated groups? Here, we investigate the opinion dynamics governed by majority rule in a minimal model comprising self-propelled agents with binary opinions performing random walks within a closed space divided by a barrier. We focus on the conditions under which initially segregated clusters of agents with opposing opinions can reach consensus. Our results reveal the existence of a critical barrier size that marks a transition between stalemate and consensus states. Near this critical point, the relaxation time to reach consensus from an initial stalemate exhibits a power-law divergence. This barrier-induced stalemate-consensus transition in a simple agent-based model offers new insights into the role of physical or social barriers in shaping opinion dynamics and social structures.