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 Sᵢ 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 (>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)= SᵢSⱼ (where r is the $i-j$ distance) decrease algebraically in a slow, non-integrable way. Specifically, we find C(r)~ r⁻¹, or C(r)~ r⁻⁽⁶⁻⁾, or C(r)~ r⁻ for >5, 3< ≤ 5 and 0≤ ≤ 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)~ t1/2 for >5, L(t)~ t5/2 for 4< ≤ 5, and L(t)~ t5/8 for 3≤ ≤ 4. For 0≤ < 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.