Majority-vote model on hyperbolic lattices

Abstract

We study the critical properties of a non-equilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents $1/\nu$, $\beta/\nu$ and $\gamma/\nu$ are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation $2\beta/\nu+\gamma/\nu=D_{\mathrm{eff}}$, where $D_{\mathrm{eff}}$ is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.