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)∝r^{-α} (α>0). In the thermodynamic limit N→∞ 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) decreases algebraically in a slow, nonintegrable way. Specifically, we find C(r)∼r^{-1}, or C(r)∼r^{-(6-α)}, or C(r)∼r^{-α} for α>5, 3<α≤5, and 0≤α≤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→∞). We find L(t)∼t^{1/2} for α>5, L(t)∼t^{5/2α} for 4<α≤5, and L(t)∼t^{5/8} for 3≤α≤4. For 0≤α<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 spatial dimension.