Ordering Kinetics of the two-dimensional voter model with long-range interactions

Abstract

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance $r$ with probability $P(r) \propto r^{-\al}$. The model is characterized by different regimes, as $\al$ is varied. For $\al > 4$ the behaviour is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L(t) \propto \sqrt{t}$, until consensus is reached in a time or order $N\ln N$, $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slow as $\rho(t) \propto 1/\ln t$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbors interactions. For $0<\al \leq 4$ standard scaling is reinstated, and the correlation length increases algebraically as $L(t)\propto t^{1/z}$, with $1/z=2/\al$ for $3<\al<4$ and $1/z=2/3$ for $0<\al<3$. In addition, for $\al \le 3$, $L(t)$ depends on $N$ at any time $t>0$. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the $N\to \infty$ limit. In finite systems consensus is reached in a time of order $N$ for any $\al <4$.