{
  "id": "1701.00513",
  "version": "v1",
  "published": "2017-01-02T20:18:49.000Z",
  "updated": "2017-01-02T20:18:49.000Z",
  "title": "Local spectral statistics of the addition of random matrices",
  "authors": [
    "Ziliang Che",
    "Benjamin Landon"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the local statistics of $H = V^* X V + U^* Y U$ where $V$ and $U$ are independent Haar-distributed unitary matrices, and $X$ and $Y$ are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size $N \\to \\infty$ under mild assumptions on $X$ and $Y$. Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when $V$ and $U$ are drawn from the orthogonal group. Our proof relies on the local law for $H$ proved by [Bao-Erd\\H{o}s-Schnelli] as well as the DBM convergence results of [L.-Sosoe-Yau].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-02T20:18:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local spectral statistics",
      "random matrices",
      "deterministic real diagonal matrices",
      "independent haar-distributed unitary matrices",
      "dbm convergence results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}