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

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 $P(r)\propto r^{-\text{al}}$. The model is characterized by different regimes, as $\text{al}$ is varied. For $\text{al}>4$ 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 $\rho (t)\propto 1\mathrm{/}\ln t$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbors interactions. For $0<\text{al}\le 4$ standard scaling is reinstated, and the correlation length increases algebraically as $L(t)\propto t^{1\mathrm{/}z}$, with $1\mathrm{/}z=2\mathrm{/}\text{al}$ for $3<\text{al}<4$ and $1\mathrm{/}z=2\mathrm{/}3$ for $0<\text{al}<3$. In addition, for $\text{al}\le 3$, $L(t)$ 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 $N\to \mathrm{\infty }$ limit. In finite systems consensus is reached in a time of order $N$ for any $\text{al}<4$.