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 $P(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$. The model is characterized by different regimes, as $\ensuremath{\alpha}$ is varied. For $\ensuremath{\alpha}>4$, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L(t)\ensuremath{\propto}\sqrt{t}$, until consensus is reached in a time of the order of $NlnN$, with $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as $\ensuremath{\rho}(t)\ensuremath{\propto}1/lnt$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For $0<\ensuremath{\alpha}\ensuremath{\le}4$, standard scaling is reinstated and the correlation length increases algebraically as $L(t)\ensuremath{\propto}{t}^{1/z}$, with $1/z=2/\ensuremath{\alpha}$ for $3<\ensuremath{\alpha}<4$ and $1/z=2/3$ for $0<\ensuremath{\alpha}<3$. In addition, for $\ensuremath{\alpha}\ensuremath{\le}3, L(t)$ 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 $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. In finite systems, consensus is reached in a time of the order of $N$ for any $\ensuremath{\alpha}<4$.

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