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

TL;DR

This paper analytically investigates the ordering dynamics and metastable states of a three-dimensional long-range voter model, revealing how the decay of correlations and coarsening behavior depend on the interaction range parameter.

Contribution

It provides a detailed analytical characterization of metastability, correlation decay, and coarsening dynamics in the long-range voter model across different interaction regimes.

Findings

Metastable states without full consensus in the thermodynamic limit.

Correlation functions decay algebraically with different exponents depending on alpha.

Coarsening dynamics exhibit power-law growth of correlation length with exponents depending on alpha.

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) $\pm 1$. The kinetics is such that each agent copies the opinion of another at distance $r$ chosen with probability $P(r) \propto r^{-\alpha}$ ($\al >0$). In the thermodynamic limit $N\to \infty$ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function $C(r)=\langle S_iS_j\rangle$ (where r is the $i-j$ distance) decrease algebraically in a slow, non-integrable way. Specifically, we find $C(r)\sim r^{-1}$, or $C(r)\sim r^{-(6-\al)}$, or $C(r)\sim r^{-\al}$ for $\al >5$, $3<\al \le 5$ and $0\le \al \le 3$, 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 $N\to \infty$). We find $L(t)\sim t^{\frac{1}{2}}$ for $\al >5$, $L(t)\sim t^{\frac{5}{2\al}}$ for $4<\al \le 5$, and $L(t)\sim t^{\frac{5}{8}}$ for $3\le \al \le 4$. For $0\le \al < 3$ 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 space dimension.