{
  "id": "2411.17827",
  "version": "v1",
  "published": "2024-11-26T19:07:10.000Z",
  "updated": "2024-11-26T19:07:10.000Z",
  "title": "Ordered random walks and the Airy line ensemble",
  "authors": [
    "Denis Denisov",
    "Will FitzGerald",
    "Vitali Wachtel"
  ],
  "comment": "34 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-26T19:07:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60K35",
      "60G40",
      "60F17"
    ],
    "keywords": [
      "airy line ensemble",
      "ordered random walks",
      "continuous-time random walks",
      "random matrix theory",
      "random walk steps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}