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 two-dimensional lattice, where agents on each vertex take the opinion of others at distance 𝑟 with probability 𝑃⁡(𝑟)∝𝑟−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)∝√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 𝜌⁡(𝑡)∝1/ln⁡𝑡. 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)∝𝑡1/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 𝑁→∞ limit. In finite systems, consensus is reached in a time of the order of 𝑁 for any 𝛼<4.