Majority-vote model on hyperbolic lattices

TL;DR

This paper investigates the critical behavior of the majority-vote model on hyperbolic lattices, revealing unique critical exponents and effective dimensionality, differing from regular lattice cases, through Monte Carlo simulations.

Contribution

It provides the first analysis of the majority-vote model on hyperbolic lattices, highlighting the impact of negative curvature on critical properties and universality class.

Findings

Critical exponents differ from regular lattice models.

Critical exponents satisfy a hyperscaling relation with an effective dimension.

Boundary nodes influence the ordering process.

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.