{
  "id": "1201.4709",
  "version": "v3",
  "published": "2012-01-23T13:23:42.000Z",
  "updated": "2012-04-13T20:55:44.000Z",
  "title": "Local behavior and hitting probabilities of the Airy1 process",
  "authors": [
    "Jeremy Quastel",
    "Daniel Remenik"
  ],
  "comment": "Expanded introduction, added Theorem 3, changed title from \"Regularity and continuum statistics of the Airy1 process\"",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We obtain a formula for the $n$-dimensional distributions of the Airy$_1$ process in terms of a Fredholm determinant on $L^2(\\rr)$, as opposed to the standard formula which involves extended kernels, on $L^2(\\{1,...,n\\}\\times\\rr)$. The formula is analogous to an earlier formula of [PS02] for the Airy$_2$ process. Using this formula we are able to prove that the Airy$_1$ process is H\\\"older continuous with exponent $\\frac12-$ and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the Airy$_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the Airy$_1$ process, analogous to that obtained in [CQR11] for the Airy$_2$ process.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-04-13T20:55:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "airy1 process",
      "local behavior",
      "hitting probabilities",
      "fredholm determinant",
      "standard formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4709Q"
    }
  }
}