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

Source: [Coarsening and metastability of the long-range voter model in three dimensions](<https://www.sota2.com/research/paper/68fa47e01e209131ce92b6f0>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T06:33:39.588Z

Authors: Federico Corberi, Salvatore dello Russo e Luca Smaldone

Published: 2024-06-17T10:13:40.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2406.11386>)

[PDF](<https://arxiv.org/pdf/2406.11386.pdf>)