Majority-vote model on spatially embedded networks: crossover from mean-field to Ising universality classes

We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which long-range connections are randomly added according to the probability, $P_{ij}\sim{r^{-\alpha}}$, where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $\alpha$ is a controlling parameter [J. M. Kleinberg, Nature 406, 845 (2000)]. Our results show that the collective behavior of this system exhibits a continuous order-disorder phase transition at a critical parameter, which is a decreasing function of the exponent $\alpha$. Precisely, considering the scaling functions and the critical exponents calculated, we conclude that the system undergoes a crossover among distinct universality classes. For $\alpha\le3$ the critical behavior is described by mean-field exponents, while for $\alpha\ge4$ it belongs to the Ising universality class. Finally, in the region where the crossover occurs, $3<\alpha<4$, the critical exponents are dependent on $\alpha$.