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

Abstract

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 $S_{i}$, 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 $P(r)\propto r^{-\alpha}$. 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 $3\le p<\infty$, 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 $p\to \infty$, a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed. Since for $ p > 3$ a closed set of differential equations cannot be found, we employed numerical simulations to address this case.