WIAS Preprint No. 3170, (2025)

Phase transitions for contact processes on one-dimensional networks


Authors

  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Lüchtrath, Lukas
    ORCID: 0000-0003-4969-806X
  • Mönch, Christian

2020 Mathematics Subject Classification

  • 60K35 05C82 91D30

Keywords

  • Gilbert graph, long-range percolation, continuum percolation, random connection model, scale-free network, SIS epidemics

DOI

10.20347/WIAS.PREPRINT.3170

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 integer indexed vertices 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 (Ménard et al., Ann. Sci. Éc. Norm. Supér., 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: First, for Gilbert graphs derived from stationary point processes on the real line 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. Second, for independent Bernoulli long-range percolation on the integers, where the edge probabilities are given via a polynomial in the edges length', we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever the power of said polynomial is large than two. Although our approach utilises the restrictive topology of the line, we believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in higher dimensions.

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