{
  "id": "1802.03834",
  "version": "v1",
  "published": "2018-02-11T22:48:32.000Z",
  "updated": "2018-02-11T22:48:32.000Z",
  "title": "Continuum directed random polymers on disordered hierarchical diamond lattices",
  "authors": [
    "Jeremy Clark"
  ],
  "comment": "25 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "I discuss a family of models for a continuum directed random polymer in a disordered environment in which the polymer lives on a fractal, $D^{b,s}$, called the \\textit{diamond hierarchical lattice}. The diamond hierarchical lattice is a compact, self-similar metric space forming a network of interweaving pathways (continuum polymers) connecting a beginning node, $A$, to a termination node, $B$. This fractal depends on a branching parameter $b\\in \\{2,3,\\cdots\\}$ and a segmenting number $s\\in \\{2,3,\\cdots\\}$, and there is a canonical uniform probability measure $\\mu$ on the collection of directed paths, $\\Gamma^{b,s}$, for which the intersection set of two randomly chosen paths is almost surely either finite or of Hausdorff dimension $(\\log s -\\log b)/\\log s$ when $s\\geq b$. In the case $s>b$, my focus is on random measures on the set of directed paths that can be formulated as a subcritcal Gaussian multiplicative chaos measure with expectation $\\mu$. When normalized, this random path measure is analogous to the continuum directed random polymer (CDRP) introduced by Alberts, Khanin, Quastel [Journal of Statistical Physics 154, 305-326 (2014)], which is formally related to the stochastic heat equation for a $(1+1)$-dimension polymer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-11T22:48:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum directed random polymer",
      "disordered hierarchical diamond lattices",
      "subcritcal gaussian multiplicative chaos measure",
      "random path measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}