{
  "id": "1603.03817",
  "version": "v1",
  "published": "2016-03-11T23:05:28.000Z",
  "updated": "2016-03-11T23:05:28.000Z",
  "title": "Eigenvalue vs perimeter in a shape theorem for self-interacting random walks",
  "authors": [
    "Marek Biskup",
    "Eviatar B. Procaccia"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study paths of time-length $t$ of a continuous-time random walk on $\\mathbb Z^2$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\\beta$; the \"energy\" is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit $t\\to\\infty$ followed by $\\beta\\to\\infty$. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in $\\mathbb R^2$, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-11T23:05:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B41",
      "60D05",
      "60F99"
    ],
    "keywords": [
      "self-interacting random walks",
      "shape theorem",
      "eigenvalue",
      "principal harmonic frequency",
      "finite edge weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303817B"
    }
  }
}