{
  "id": "2009.11176",
  "version": "v1",
  "published": "2020-09-23T14:39:13.000Z",
  "updated": "2020-09-23T14:39:13.000Z",
  "title": "Edge scaling limit of Dyson Brownian motion at equilibrium for general $β\\geq 1$",
  "authors": [
    "Benjamin Landon"
  ],
  "comment": "32 Pages. Draft, comments welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For general $\\beta \\geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \\to \\infty$. For each fixed time, this ensemble is distributed as the Airy$_\\beta$ random point field. We prove that the increments of the limiting process are locally Brownian. When $\\beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{\\\"o}lder for any $r<1-(1+\\beta)^{-1}$. Furthermore for all $\\beta \\geq 1$ we show that the limiting process solves an SDE in a weak sense. When $\\beta=2$ this limiting process is the Airy line ensemble.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-23T14:39:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dyson brownian motion",
      "edge scaling limit",
      "limiting process",
      "equilibrium",
      "random point field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}