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) +/- 1. The kinetics is such that each agent copies the opinion of another at distance r chosen with probability P(r) proportional to r(-alpha) (a > 0). In the thermodynamic limit N ->infinity the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function C(r) = < SiSj > (where r is the i-j distance) decrease algebraically in a slow, non-integrable way. Specifically, we find C(r) similar to r(-1), or C(r)similar to r(6-a), or C(r)similar to r(-a) for a >5, 3< a <= 5 and 0 <= <= a <= 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 ->infinity). We find L(t)similar to t1/2 for a <= 5 L(t)similar to t5{2\al}}$ for $4<\al \le 5$, and L(t)similar to t58 for $3\le \al \le 4$. For $0\le \al < 3$ 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.