Coarsening kinetics in 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, $J(r)\propto r^{-\alpha }$. 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 $p=1$ and $p=2$, while for $p\ge 3$ it falls into the universality class of the Ising model.