From Checkerboard to 4.5%: Emergence and Phase Transitions in a Ternary Majority Vote Model

This study explores a simple ternary (-1, 0, +1) majority vote model. Starting from a checkerboard initialization (+1 where i+j is even, -1 otherwise), the system spontaneously emerges a stable zero-state fraction of approximately 4.5% without any pre-specified global rules, and this fraction is independent of system size. Through systematic parameter scanning, we discover two continuous phase transitions: (1) Percolation transition driven by self-protection attraction between zero-states, critical point ≈ 0.99, critical exponent β = 0.447, belonging to the mean-field percolation universality class; (2) Gravitational transition driven by long-range gravitational attraction, critical point ≈ 1.2, critical exponent β = 0.85, potentially representing a new universality class. In a cosmological analogy, the 4.5% zero-state corresponds to ordinary matter (∼5% of the universe), the additional 25-40% aggregation corresponds to dark matter (∼27%), and the remaining ±1 corresponds to dark energy (∼68%).