{
  "abstract": "We consider $\\nablaφ$ height functions with even and convex interaction energy $W$ on the lattice $\\mathbb{Z}^d$, which are restricted to take values in the set $\\{-S, \\ldots, S\\}$ for some integer $S \\ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\\in[1,2]$. (c) Dimensions $d\\ge 3$ at both low and high temperatures, $T\\in (0,\\frac{W(1)d}{4(\\ln d+\\ln 8)})\\cup(d W(2S),\\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\\frac{1}{2},\\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\\{0,1,\\ldots\\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\\{-S+\\frac{1}{2}, \\ldots, S-\\frac{1}{2}\\}$, $S \\ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont--El Mellouki--Fröhlich (1986).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07578",
  "author_total": 2,
  "authors": [
    {
      "name": "Max Mihailescu"
    },
    {
      "name": "Ron Peled"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5s4ebbsekr95r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07578",
  "fulltext_url": null,
  "id": "5s4ebbsekr95r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07578",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212022885",
  "pdf_url": "https://arxiv.org/pdf/2609.07578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5s4ebbsekr95r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.07578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Entropic repulsion to the middle layer",
  "title_text": "Entropic repulsion to the middle layer",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5s4ebbsekr95r",
  "venue": "arXiv",
  "versions": []
}