{
  "id": "2306.00896",
  "version": "v1",
  "published": "2023-06-01T16:54:57.000Z",
  "updated": "2023-06-01T16:54:57.000Z",
  "title": "Boundary conditions and universal finite-size scaling for the hierarchical $|\\varphi|^4$ model in dimensions 4 and higher",
  "authors": [
    "Emmanuel Michta",
    "Jiwoon Park",
    "Gordon Slade"
  ],
  "comment": "102 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|\\varphi|^4$ model for all integers $n \\ge 1$ in all dimensions $d\\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that for a volume of size $R^{d}$ with periodic boundary conditions the infinite-volume critical point is an effective finite-volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order $R^{-2}$. For both boundary conditions, the average field has the same non-Gaussian limit within a critical window of width $R^{-d/2}$ around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount $R^{-2}$. In particular, at the infinite-volume critical point the susceptibility scales as $R^{d/2}$ for periodic boundary conditions and as $R^{2}$ for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non-hierarchical) models on $\\mathbb{Z}^d$ in dimensions $d \\ge 4$. For $d=4$ we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-01T16:54:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B27",
      "82B28",
      "60K35"
    ],
    "keywords": [
      "periodic boundary conditions",
      "free boundary conditions",
      "universal finite-size scaling",
      "effective critical point",
      "dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 102,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}