Ordering kinetics with long-range interactions: interpolating between voter and Ising models

We study the ordering kinetics of a generalization of the voter model with long-range interactions, the p-voter model, in one dimension. It is defined in terms of boolean variables Sᵢ, agents or spins, located on sites i of a lattice, each of which takes in an elementary move the state of the majority of p other agents at distances r chosen with probability P(r)∝ r-α. For $p=2$ the model can be exactly mapped onto the case with $p=1$, which amounts to the voter model with long-range interactions decaying algebraically. For 3≤ p<∞, instead, the dynamics falls into the universality class of the one-dimensional Ising model with long-ranged coupling constant $J(r)=P(r)$ quenched to small finite temperatures. In the limit p→ ∞, a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed.