{
  "id": "2301.04886",
  "version": "v1",
  "published": "2023-01-12T09:01:58.000Z",
  "updated": "2023-01-12T09:01:58.000Z",
  "title": "Random vectors on the spin configuration of a Curie-Weiss model on Erdős-Rényi random graphs",
  "authors": [
    "Dominik R. Bach"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erd\\H{o}s-R\\'enyi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If $Np\\longrightarrow\\infty$ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for $\\beta<1$ in the regime $Np\\longrightarrow\\infty$. We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-12T09:01:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random vectors",
      "erdős-rényi random graphs",
      "spin configuration",
      "central limit theorem",
      "large numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}