Kinetics of the one-dimensional voter model with long-range interactions

Abstract

The one-dimensional long-range voter model, where an agent takes the opinion of another at distance $r$ with probability $\propto r^{-\alpha}$, is studied analytically. The model displays rich and diverse features as $\alpha$ is changed. For $\alpha >3$ the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as $R(t)\propto t^{1/2}$ until consensus (a fully ordered configuration) is reached. The correlation function $C(r,t)$ between two agents at distance $r$ obeys dynamical scaling with sizeable corrections at large distances $r>r^*(t)$, slowly fading away in time. For $2< \alpha \le 3$ violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as $t^{(\alpha-2)/(\alpha-1)}$, and the distance $L(t)\propto t^{1/(\alpha-1)}$ over which correlations extend. For $\alpha \le 2$ the system reaches a partially ordered stationary state, characterised by an algebraic correlator, % $C(r)\propto r^{-(2-\alpha)}$, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of $N$ sites, global consensus is achieved after a time $T \propto N^2$ for $\alpha>3$, $T \propto N^{\alpha-1}$ for $2<\alpha \le 3$, and $T \propto N$ for $\alpha \le 2$.