{
  "id": "2406.11386",
  "version": "v1",
  "published": "2024-06-17T10:13:40.000Z",
  "updated": "2024-06-17T10:13:40.000Z",
  "title": "Coarsening and metastability of the long-range voter model in three dimensions",
  "authors": [
    "Federico Corberi",
    "Salvatore dello Russo e Luca Smaldone"
  ],
  "comment": "17 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-17T10:13:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coarsening",
      "three-dimensional long-range voter model",
      "equal-time correlation function",
      "generic space dimension",
      "finite system metastability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}