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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, $P_{ij}\sim{r^{-α}}$, where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $α$ 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 $α$. Precisely, considering the scaling functions and the critical exponents calculated, we conclude that the system undergoes a crossover among distinct universality classes. For $α\le3$ the critical behavior is described by mean-field exponents, while for $α\ge4$ it belongs to the Ising universality class. Finally, in the region where the crossover occurs, $3<α<4$, the critical exponents are dependent on $α$.