We investigate the dynamics of the three-state majority-vote model on a simple cubic lattice. In the model, each site adopts the majority opinion of its neighbors with probability $1-q$, and a different opinion with probability $q$ (the noise parameter). We monitor the magnetization, the magnetic susceptibility, and the fourth-order Binder cumulant using Monte Carlo simulations. The magnetization shows a rapid change near the critical noise parameter $q_c$, consistent with the behavior expected at a weak first-order phase transition. The critical exponent ratio $\gamma /\nu =2.91\left(12\right)$, close to three, is obtained from the peaks in the magnetic susceptibility. Scaling analysis of the susceptibility confirms $\gamma /\nu =2.91$ and yields the correlation-length exponent $\nu \approx 1/3$ corresponding to the weak first-order phase transition. Additional evidence for the weak first-order transition is provided by the Binder cumulant, which develops a pronounced negative dip near $q_c$ for large system size. These results demonstrate that the three-state majority-vote model on a simple cubic lattice undergoes a weak first-order phase transition.

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• Critical phenomena
• Order parameters
• Phase transitions
• Phase transitions by order
• Transition temperature
• Lattice models in statistical physics
• Nonequilibrium lattice models
• Finite-size scaling
• Metropolis algorithm