{
  "id": "2501.16858",
  "version": "v2",
  "published": "2025-01-28T11:14:09.000Z",
  "updated": "2025-08-05T00:22:42.000Z",
  "title": "Phase transitions for contact processes on one-dimensional networks",
  "authors": [
    "Benedikt Jahnel",
    "Lukas Lüchtrath",
    "Christian Mönch"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by the integers that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (M\\'enard et al., Ann. Sci. \\'Ec. Norm. Sup\\'er., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: Firstly, for Gilbert graphs derived from stationary point processes on $\\mathbb{R}$ marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Secondly, for independent Bernoulli long-range percolation on $\\mathbb{Z}$, with coupling constants $J_{x,y}\\asymp |x-y|^{-\\delta}$, we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever $\\delta>2$. We believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in dimensions $d > 1$ as long as the degree distribution of the underlying network has at least finite $d$-th moment. We support this by proving that no phase transition exists if the $d$-th moment is infinite.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-08-05T00:22:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "05C82",
      "91D30"
    ],
    "keywords": [
      "contact processes",
      "one-dimensional networks",
      "degree distribution",
      "th moment",
      "scale-free random geometric graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}