Coarsening and metastability of the long-range voter model in three dimensions

We study analytically the ordering kinetics and the final metastable states in the three-dimensional long-range voter model where $N$ agents described by a boolean spin variable $S_i$ can be found in two states (or opinion) $\pm 1$. The kinetics is such that each agent copies the opinion of another at distance $r$ chosen with probability $P(r)\propto r^{-\alpha }$ ($\text{al}>0$). In the thermodynamic limit $N\to \mathrm{\infty }$ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function $C(r)=⟨S_i{S}_j⟩$ (where r is the $i-j$ distance) decrease algebraically in a slow, non-integrable way. Specifically, we find $C(r)\sim r^{-1}$, or $C(r)\sim r^{-(6-\text{al})}$, or $C(r)\sim r^{-\text{al}}$ for $\text{al}>5$, $3<\text{al}\le 5$ and $0\le \text{al}\le 3$, respectively. In a finite system metastability is escaped after a time of order $N$ and full ordering is eventually achieved. The dynamics leading to metastability is of the coarsening type, with an ever increasing correlation length $L(t)$ (for $N\to \mathrm{\infty }$). We find $L(t)\sim t^{\frac{1}{2}}$ for $\text{al}>5$, $L(t)\sim t^{\frac{5}{2\text{al}}}$ for $4<\text{al}\le 5$, and $L(t)\sim t^{\frac{5}{8}}$ for $3\le \text{al}\le 4$. For $0\le \text{al}<3$ there is not macroscopic coarsening because stationarity is reached in a microscopic time. Such results allow us to conjecture the behavior of the model for generic space dimension.