{
  "id": "2302.11379",
  "version": "v1",
  "published": "2023-02-22T13:52:51.000Z",
  "updated": "2023-02-22T13:52:51.000Z",
  "title": "From stability to chaos in last-passage percolation",
  "authors": [
    "Daniel Ahlberg",
    "Maria Deijfen",
    "Matteo Sfragara"
  ],
  "comment": "11 pages. A video summary may be found at https://youtu.be/Y29t_KUzv7k",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the transition from stability to chaos in a dynamic last passage percolation model on $\\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the square $[0,n]^2$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0)$ to $(n,n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \\asymp \\frac{\\mathrm{Var}(T)}{n}$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-22T13:52:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "last-passage percolation",
      "passage time",
      "initial weight configuration",
      "passage percolation model",
      "transition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}