{
  "id": "1712.03881",
  "version": "v1",
  "published": "2017-12-11T16:45:41.000Z",
  "updated": "2017-12-11T16:45:41.000Z",
  "title": "Edge statistics of Dyson Brownian motion",
  "authors": [
    "Benjamin Landon",
    "Horng-Tzer Yau"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $\\eta_* \\geq N^{-2/3}$ and no outliers, then after times $t \\gg \\sqrt{ \\eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{\\varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-Erd\\H{o}s-Yau-Yin] and the energy estimate of [Bourgade-Erd\\H{o}s-Yau].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-11T16:45:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dyson brownian motion",
      "edge statistics",
      "eigenvalue rigidity results similar",
      "rough square root behavior",
      "main result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}