{
  "id": "2312.07726",
  "version": "v1",
  "published": "2023-12-12T20:42:13.000Z",
  "updated": "2023-12-12T20:42:13.000Z",
  "title": "Optimal lower bound for the variance of hitting times for simple random walks on graphs",
  "authors": [
    "Rafael Chiclana",
    "Yuval Peres"
  ],
  "comment": "6 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a graph with $n$ vertices, we prove that the variance of the hitting time from a vertex $x$ to a vertex $y$, denoted $\\tau_y$, is at least of the order $\\mathbb{E}_x(\\tau_y)^2 / \\log n$. When the graph is a tree, we show that $n$ can be replaced by the graph's distance between vertices $x$ and $y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-12T20:42:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C81",
      "05C05"
    ],
    "keywords": [
      "simple random walk",
      "hitting time",
      "optimal lower bound",
      "reach specific target vertices",
      "sharp lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}