{
  "id": "2312.03300",
  "version": "v1",
  "published": "2023-12-06T05:54:34.000Z",
  "updated": "2023-12-06T05:54:34.000Z",
  "title": "Non-backtracking eigenvector delocalization for random regular graphs",
  "authors": [
    "Xiangyi Zhu",
    "Yizhe Zhu"
  ],
  "comment": "11 pages, 5 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this paper, we take the first step in analyzing its bulk eigenvector behaviors. We demonstrate that for the non-backtracking operator $B$ of a random $d$-regular graph, its eigenvectors corresponding to nontrivial eigenvalues are completely delocalized with high probability. Additionally, we show complete delocalization for a reduced $2n \\times 2n$ non-backtracking matrix $\\widetilde{B}$. By projecting all eigenvalues of $\\widetilde{B}$ onto the real line, we obtain an empirical measure that converges weakly in probability to the Kesten-McKay law for fixed $d\\geq 3$ and to a semicircle law as $d \\to\\infty$ with $n \\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-06T05:54:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random regular graphs",
      "non-backtracking eigenvector delocalization",
      "non-backtracking operator",
      "random matrix theory",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}