{
  "id": "2210.13394",
  "version": "v1",
  "published": "2022-10-24T16:48:55.000Z",
  "updated": "2022-10-24T16:48:55.000Z",
  "title": "Existence of a tricritical point for the Blume-Capel model on $\\mathbb{Z}^d$",
  "authors": [
    "Trishen Gunaratnam",
    "Dmitrii Krachun",
    "Christoforos Panagiotis"
  ],
  "comment": "41 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove the existence of a tricritical point for the Blume-Capel model on $\\mathbb{Z}^d$ for every $d\\geq 2$. The proof in $d\\geq 3$ relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In $d=2$, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in $d=2$, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any $d\\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-24T16:48:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "blume-capel model",
      "tricritical point",
      "dilute random cluster model extend",
      "dilute random cluster representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}