Ordering kinetics of the two-dimensional voter model with long-range interactions

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a twodimensional lattice, where agents on each vertex take the opinion of others at distance r with probability P(r) oc r-alpha. The model is characterized by different regimes, as alpha is varied. For alpha > 4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of atypical size growing as L(t) oc ./t, until consensus is reached in a time of the order of N ln N, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as p(t) oc 1/ ln t. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0 < alpha 4, standard scaling is reinstated and the correlation length increases algebraically as L(t) oc t1/z, with 1/z = 2/alpha for 3 < alpha < 4 and 1/z = 2/3 for 0 < alpha < 3. In addition, for alpha 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 N oo limit. In finite systems, consensus is reached in a time of the order of N for any alpha < 4.