Noisy multistate voter model for flocking in finite dimensions

We study a model for the collective behavior of self-propelled particles subject to pairwise copying interactions and noise. Particles move at a constant speed $v$ on a two--dimensional space and, in a single step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle, with the addition of a perturbation of amplitude $\eta$ (noise). We investigate how the global level of particles' alignment (order) is affected by their motion and the noise amplitude $\eta$. In the static case scenario $v=0$ where particles are fixed at the sites of a square lattice and interact with their first neighbors, we find that for any noise $\eta_c>0$ the system reaches a steady state of complete disorder in the thermodynamic limit, while for $\eta=0$ full order is eventually achieved for a system with any number of particles $N$. Therefore, the model displays a transition at zero noise when particles are static, and thus there are no ordered steady states for a finite noise ($\eta>0$). We show that the finite-size transition noise vanishes with $N$ as $\eta_c^{1D} \sim N^{-1}$ and $\eta_c^{2D} \sim \left(N \ln N \right)^{-1/2}$ in one and two--dimensional lattices, respectively, which is linked to known results on the behavior of a type of noisy voter model for catalytic reactions. When particles are allowed to move in the space at a finite speed $v>0$, an ordered phase emerges, characterized by a fraction of particles moving in a similar direction. The system exhibits an order-disorder phase transition at a noise amplitude $\eta_c>0$ that is proportional to $v$, and that scales approximately as $\eta_c \sim v \, (-\ln v)^{-1/2}$ for $v \ll 1$. These results show that the motion of particles is able to sustain a state of global order in a system with voter-like interactions.