{
  "id": "2301.07086",
  "version": "v1",
  "published": "2023-01-17T18:52:06.000Z",
  "updated": "2023-01-17T18:52:06.000Z",
  "title": "A continuum limit for the Laplace operator on metric graphs",
  "authors": [
    "Sidney Holden",
    "Geoffrey Vasil"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Metric graphs -- with continuous real-valued edges -- have been studied extensively as domains for solutions of PDEs. We study the action of the Laplace operator on each edge as a metric graph fills in a $n-$dimensional Riemannian manifold, e.g., a very dense spider web in the planar disc or ball of wire within the $2-$sphere. We derive an $n$-dimensional limiting differential operator whose low-order eigenmodes approximate the low-order eigenmodes of the metric graph Laplace operator. High-density metric graphs exhibit nontrivial, continuum-like behaviour at low frequencies and purely graph-like behaviour at high frequencies. In the continuum limit, we find the emergence of a new symmetric tensor field but with an unusual distance scaling compared to the traditional Riemannian metric. The limiting operator from a metric graph implies a possible new kind of continuous geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-17T18:52:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum limit",
      "metric graph laplace operator",
      "traditional riemannian metric",
      "dimensional riemannian manifold",
      "dense spider web"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}