Three-state majority-vote model on a simple cubic lattice

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 γ/ν=2.91(12), close to three, is obtained from the peaks in the magnetic susceptibility. Scaling analysis of the susceptibility confirms γ/ν=2.91 and yields the correlation-length exponent ν≈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.