{
  "id": "1610.00198",
  "version": "v1",
  "published": "2016-10-01T22:04:54.000Z",
  "updated": "2016-10-01T22:04:54.000Z",
  "title": "Expected Depth of Random Walks on Groups",
  "authors": [
    "Khalid bou-Rabee",
    "Ioan Manolescu",
    "Aglaia Myropolska"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GR",
    "math.PR"
  ],
  "abstract": "For $G$ a finitely generated group and $g \\in G$, we say $g$ is detected by a normal subgroup $N \\lhd G$ if $g \\notin N$. The depth $D_G(g)$ of $g$ is the lowest index of a normal, finite index subgroup $N$ that detects $g$. In this paper we study the expected depth, $\\mathbb E[D_G(X_n)]$, where $X_n$ is a random walk on $G$. We give several criteria that imply that $$\\mathbb E[D_G(X_n)] \\xrightarrow[n\\to \\infty]{} 2 + \\sum_{k \\geq 2}\\frac{1}{[G:\\Lambda_k]}\\, ,$$ where $\\Lambda_k$ is the intersection of all normal subgroups of index at most $k$. We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-01T22:04:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65"
    ],
    "keywords": [
      "random walk",
      "expected depth",
      "normal subgroup",
      "finite index subgroup",
      "lowest index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}