2025/01/28 by Jahnel, Benedikt, Lüchtrath, Lukas, Mönch, Christian · 2 citations
#05C82 #60K35 (primary) #91D30 (secondary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2501.16858
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 (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: Firstly, for Gilbert graphs derived from stationary point processes on 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 ℤ, with coupling constants Jx,y\asymp |x-y|-δ, we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever δ>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.