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 $r>r^{\ast }(t)$, 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 $\alpha \le 2$.

16 pages, 9 figures

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• Statistical physics of social dynamics
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• Langevin description of critical phenomena with two symmetric absorbing states
• Kinetics of the Two-dimensional Long-range Ising Model at Low Temperatures
• Ordering Kinetics of the two-dimensional voter model with long-range interactions
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Cited by in corpus (5)

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• Physical ageing from generalised time-translation-invariance
• Ordering kinetics with long-range interactions: interpolating between voter and Ising models
• Coarsening in the Persistent Voter Model: analytical results
• Schrödinger-invariance in the voter model