Coarsening kinetics in one-dimensional spin systems with long-range interactions: from voter to Ising

In this paper, we start reviewing the main features of the one-dimensional Ising model with long-range interactions, where the spin-spin coupling decays as a power law, J(r) ∝ r-α J ( r ) ∝ r - α . We then discuss the key properties of the one-dimensional voter model, in which two agents (spins) at distance r interact with a power-law probability with the same form of J ( r ). The two models are compared, and the so-called p -voter model is presented, which provides a framework to interpolate between them. Specifically, the p -voter model reduces to the voter model for $$p = 1$$ p = 1 and $$p = 2$$ p = 2 , while for p ≥ 3 p ≥ 3 it falls into the universality class of the Ising model.