{
  "id": "2112.05133",
  "version": "v2",
  "published": "2021-12-09T18:59:21.000Z",
  "updated": "2023-07-31T17:16:09.000Z",
  "title": "Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor",
  "authors": [
    "Reza Gheissari",
    "Eyal Lubetzky"
  ],
  "comment": "34 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the interface of the Ising model in a box of side-length $n$ in $\\mathbb Z^3$ at low temperature $1/\\beta$ under Dobrushin's boundary conditions, conditioned to stay in a half-space above height $h$ (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height $0$. With the floor, for small $h$, the model is expected to exhibit {\\it entropic repulsion}, where the typical height of the interface lifts off of $0$. Detailed understanding of the SOS model -- a more tractable height function approximation of 3D Ising -- due to Caputo et al., suggests that there is a single integer value $-h_n^* \\sim -c\\log n$ of the floor height, delineating the transition between rigidity at height $0$ and entropic repulsion. We identify an explicit $h_n^*=( c_\\star+o(1))\\log n$ such that, for the typical Ising interface above a hard floor at $h$, all but an $\\epsilon(\\beta)$-fraction of the sites are propelled to be above height $0$ if $h < h_n^*-1$, whereas all but an $\\epsilon(\\beta)$-fraction of the sites remain at height $0$ if $h\\geq h_n^*$. Further, $c_\\star$ is such that the typical height of the unconditional maximum is $(2c_\\star + o(1))\\log n$; this confirms scaling predictions from the SOS approximation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-07-31T17:16:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B24",
      "82B41"
    ],
    "keywords": [
      "entropic repulsion",
      "3d ising interfaces",
      "hard floor",
      "tractable height function approximation",
      "dobrushins boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}