{
  "id": "2110.03808",
  "version": "v2",
  "published": "2021-10-07T22:06:18.000Z",
  "updated": "2022-01-12T13:47:09.000Z",
  "title": "Diffusive scaling limit of the Busemann process in Last Passage Percolation",
  "authors": [
    "Ofer Busani"
  ],
  "comment": "69 p. some results that were in the body of the text were moved into the Main Results section, references were added",
  "categories": [
    "math.PR"
  ],
  "abstract": "In exponential last passage percolation, we consider the rescaled Busemann process $x\\mapsto N^{-1/3}B^\\rho_{0,[xN^{2/3}]e_1} \\,\\, (x\\in\\mathbb{R})$, as a process parametrized by the scaled density $\\rho=1/2+\\frac{\\mu}{4} N^{-1/3}$, and taking values in $C(\\mathbb{R})$. We show that these processes, as $N\\rightarrow \\infty$, have a c\\`adl\\`ag scaling limit $G=(G_\\mu)_{\\mu\\in \\mathbb{R}}$, parametrized by $\\mu$ and taking values in $C(\\mathbb{R})$. The limiting process $G$, which can be thought of as the Busemann process under the KPZ scaling, can be described as an ensemble of \"sticky\" lines of Brownian regularity. We believe $G$ is the universal scaling limit of Busemann processes in the KPZ universality class. Our proof provides insight into this limiting behaviour by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Sepp\\\"al\\\"ainen in arXiv:1808.09069, and a sorting algorithm of random walks introduced by O'Connell and Yor [32]",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-12T13:47:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "passage percolation",
      "diffusive scaling limit",
      "kpz universality class",
      "rescaled busemann process",
      "brownian regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}