{
  "id": "2302.11367",
  "version": "v1",
  "published": "2023-02-22T13:35:52.000Z",
  "updated": "2023-02-22T13:35:52.000Z",
  "title": "Chaos, concentration and multiple valleys in first-passage percolation",
  "authors": [
    "Daniel Ahlberg",
    "Maria Deijfen",
    "Matteo Sfragara"
  ],
  "comment": "30 pages, 2 figures. A video summary may be found at https://youtu.be/Y29t_KUzv7k",
  "categories": [
    "math.PR"
  ],
  "abstract": "A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations and a chaotic behaviour of the ground state in certain Gaussian disordered systems. The purpose of this paper is to show that Chatterjee's work gives evidence of a more general principle, by establishing an analogous connection between fluctuations and chaos in the context of first-passage percolation. The notion of `chaos' here refers to the sensitivity of the time-minimising path between two points when exposed to a slight perturbation. More precisely, we resample a small proportion of the edge weights, and find that a vanishing fraction of the edges on the distance-minimising path still belongs to the time-minimising path obtained after resampling. We also show that the chaotic behaviour implies the existence of a large number of almost-optimal paths that are almost disjoint from the time-minimising path, a phenomenon known as `multiple valleys'.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-22T13:35:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "first-passage percolation",
      "multiple valleys",
      "time-minimising path",
      "concentration",
      "chaotic behaviour implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}