{
  "id": "2312.16008",
  "version": "v1",
  "published": "2023-12-26T11:27:12.000Z",
  "updated": "2023-12-26T11:27:12.000Z",
  "title": "Potts and random cluster measures on locally regular-tree-like graphs",
  "authors": [
    "Anirban Basak",
    "Amir Dembo",
    "Allan Sly"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Fixing $\\beta \\ge 0$ and an integer $q \\ge 2$, consider the ferromagnetic $q$-Potts measures $\\mu_n^{\\beta,B}$ on finite graphs ${\\sf G}_n$ on $n$ vertices, with external field strength $B \\ge 0$ and the corresponding random cluster measures $\\varphi^{q,\\beta,B}_{n}$. Suppose that as $n \\to \\infty$ the uniformly sparse graphs ${\\sf G}_n$ converge locally to an infinite $d$-regular tree ${\\sf T}_{d}$, $d \\ge 3$. We show that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which has been proved in case $d$ is even, or when $B=0$), yields the local weak convergence of $\\varphi^{q,\\beta,B}_n$ and $\\mu_n^{\\beta,B}$ to the corresponding free or wired random cluster measure, Potts measure, respectively, on ${\\sf T}_{d}$. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing {as limit points on} the critical line $\\beta_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $\\beta>\\beta_c$, we further establish a pure-state decomposition by showing that conditionally on the same dominant color $1 \\le k \\le q$, the $q$-Potts measures on such edge-expander graphs ${\\sf G}_n$ converge locally to the $q$-Potts measure on ${\\sf T}_{d}$ with a boundary wired at color $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-26T11:27:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B26"
    ],
    "keywords": [
      "random cluster measure",
      "locally regular-tree-like graphs",
      "potts measure",
      "larger potts bethe functional value",
      "bethe replica symmetric prediction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}