{
  "id": "2102.09100",
  "version": "v1",
  "published": "2021-02-18T01:17:58.000Z",
  "updated": "2021-02-18T01:17:58.000Z",
  "title": "Regularity method and large deviation principles for the Erdős--Rényi hypergraph",
  "authors": [
    "Nicholas A. Cook",
    "Amir Dembo",
    "Huy Tuan Pham"
  ],
  "comment": "37 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We develop a quantitative large deviations theory for random Bernoulli tensors. The large deviation principles rest on a decomposition theorem for arbitrary tensors outside a set of tiny measure, in terms of a novel family of norms generalizing the cut norm. Combined with associated counting lemmas, these yield sharp asymptotics for upper tails of homomorphism counts in the $r$-uniform Erd\\H{o}s--R\\'enyi hypergraph for any fixed $r\\ge 2$, generalizing and improving on previous results for the Erd\\H{o}s--R\\'enyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields (joint) upper and lower tail asymptotics for other nonlinear functionals of interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-18T01:17:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60C05",
      "60B20",
      "05C65"
    ],
    "keywords": [
      "erdős-rényi hypergraph",
      "regularity method",
      "large deviation principles rest",
      "random bernoulli tensors",
      "lower tail asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}