{
  "id": "1901.05948",
  "version": "v1",
  "published": "2019-01-17T18:44:33.000Z",
  "updated": "2019-01-17T18:44:33.000Z",
  "title": "Tail bounds for gaps between eigenvalues of sparse random matrices",
  "authors": [
    "Patrick Lopatto",
    "Kyle Luh"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu. As an application of our tail bounds, we show that for sparse Erd\\H{o}s--R\\'enyi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-17T18:44:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "sparse random matrices",
      "tail bounds",
      "first eigenvalue repulsion bound",
      "strong nodal domains",
      "error probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}