RESUMEN
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)âr^{-α} (α>0). In the thermodynamic limit Nâ∞ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function C(r)=ãS_{i}S_{j}ã (where r is the i-j distance) decreases algebraically in a slow, nonintegrable way. Specifically, we find C(r)â¼r^{-1}, or C(r)â¼r^{-(6-α)}, or C(r)â¼r^{-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 Nâ∞). We find L(t)â¼t^{1/2} for α>5, L(t)â¼t^{5/2α} for 4<α≤5, and L(t)â¼t^{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.
RESUMEN
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)âr^{-α}. The model is characterized by different regimes, as α is varied. For α>4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as L(t)âsqrt[t], until consensus is reached in a time of the order of NlnN, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as ρ(t)â1/lnt. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0<α≤4, standard scaling is reinstated and the correlation length increases algebraically as L(t)ât^{1/z}, with 1/z=2/α for 3<α<4 and 1/z=2/3 for 0<α<3. In addition, for α≤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 Nâ∞ limit. In finite systems, consensus is reached in a time of the order of N for any α<4.