{
  "id": "0709.1659",
  "version": "v1",
  "published": "2007-09-11T15:25:27.000Z",
  "updated": "2007-09-11T15:25:27.000Z",
  "title": "On the localized phase of a copolymer in an emulsion: supercritical percolation regime",
  "authors": [
    "Frank den Hollander",
    "Nicolas Pétrélis"
  ],
  "comment": "43 pages and 10 figures",
  "doi": "10.1007/s00220-008-0679-y",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \\in (0,1)$. The copolymer in the emulsion has an energy that is minus $\\alpha$ times the number of $AA$-matches minus $\\beta$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\\{(\\alpha,\\beta)\\in\\R^2\\colon \\alpha\\geq |\\beta|\\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \\cite{dHW06}, it was found that in the supercritical percolation regime $p \\geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-11T15:25:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60K37",
      "82B27"
    ],
    "keywords": [
      "supercritical percolation regime",
      "localized phase",
      "directed self-avoiding walk model",
      "free energy",
      "phase transition"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2009,
      "month": "Feb",
      "volume": 285,
      "number": 3,
      "pages": 825
    },
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009CMaPh.285..825D"
    }
  }
}