Coarsening and metastability of the long-range voter model in three dimensions

Abstract

We study analytically the ordering kinetics and the final metastable states in the three-dimensional long-range voter model where $N$ agents described by a Boolean spin variable ${S}_{i}$ can be found in two states (or opinion) $\ifmmode\pm\else\textpm\fi{}1$. The kinetics is such that each agent copies the opinion of another at distance $r$ chosen with probability $P(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}>0$). In the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function $C(r)=\ensuremath{\langle}{S}_{i}{S}_{j}\ensuremath{\rangle}$ (where $r$ is the $i\ensuremath{-}j$ distance) decreases algebraically in a slow, nonintegrable way. Specifically, we find $C(r)\ensuremath{\sim}{r}^{\ensuremath{-}1}$, or $C(r)\ensuremath{\sim}{r}^{\ensuremath{-}(6\ensuremath{-}\ensuremath{\alpha})}$, or $C(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\alpha}>5, 3<\ensuremath{\alpha}\ensuremath{\le}5$, and $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$, respectively. In a finite system metastability is escaped after a time of order $N$ and full ordering is eventually achieved. The dynamics leading to metastability is of the coarsening type, with an ever-increasing correlation length $L(t)$ (for $N\ensuremath{\rightarrow}\ensuremath{\infty}$). We find $L(t)\ensuremath{\sim}{t}^{\frac{1}{2}}$ for $\ensuremath{\alpha}>5, L(t)\ensuremath{\sim}{t}^{\frac{5}{2\ensuremath{\alpha}}}$ for $4<\ensuremath{\alpha}\ensuremath{\le}5$, and $L(t)\ensuremath{\sim}{t}^{\frac{5}{8}}$ for $3\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$. For $0\ensuremath{\le}\ensuremath{\alpha}<3$ there is not macroscopic coarsening because stationarity is reached in a microscopic time. Such results allow us to conjecture the behavior of the model for generic spatial dimension.

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