{
  "id": "1712.03736",
  "version": "v1",
  "published": "2017-12-11T12:05:37.000Z",
  "updated": "2017-12-11T12:05:37.000Z",
  "title": "Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy",
  "authors": [
    "Hubert Lacoin"
  ],
  "comment": "55 pages, 3 Figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $(\\phi(x))_{x\\in \\mathbb Z^2}$, and the energy functional $$V(\\phi)=\\beta \\sum_{x\\sim y}|\\phi(x)-\\phi(y)|-\\sum_{x}\\left( h{\\bf 1}_{\\{\\phi(x)=0\\}}-\\infty{\\bf 1}_{\\{\\phi(x)<0\\}} \\right).$$ We prove that for $\\beta$ sufficiently large, there exists a decreasing sequence $(h^*_n(\\beta))_{n\\ge 0}$, satisfying $\\lim_{n\\to\\infty}h^*_n(\\beta)=h_w(\\beta),$ and such that: $(A)$ The free energy associated with the system is infinitely differentiable on $\\mathbb R \\setminus \\left(\\{h^*_n\\}_{n\\ge 1}\\cup h_w(\\beta)\\right)$, and not differentiable on $\\{h^*_n\\}_{n\\ge 1}$. $(B)$ For each $n\\ge 0$ within the interval $(h^*_{n+1},h^*_n)$ (with the convention $h^*_0=\\infty$), there exists a unique translation invariant Gibbs state which is localized around height $n$, while at a point of non-differentiability, at least two ergodic Gibbs state coexist. The respective typical heights of these two Gibbs states are $n-1$ and $n$. The value $h^*_n$ corresponds thus to a first order layering transition from level $n$ to level $n-1$. These results combined with those obtained in [23] provide a complete description of the wetting and layering transition for SOS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-11T12:05:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free energy",
      "unique translation invariant gibbs state",
      "solid-on-solid",
      "regularity",
      "ergodic gibbs state coexist"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}