Phase transitions for contact processes on one-dimensional networks
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arXiv
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| Natura: | Preprint |
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2025
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| author | Jahnel, Benedikt Lüchtrath, Lukas Mönch, Christian |
| author_facet | Jahnel, Benedikt Lüchtrath, Lukas Mönch, Christian |
| contents | 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 $\mathbb{R}$ 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 $J_{x,y}\asymp |x-y|^{-δ}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transitions for contact processes on one-dimensional networks Jahnel, Benedikt Lüchtrath, Lukas Mönch, Christian Probability 60K35 (primary), 05C82, 91D30 (secondary) 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 $\mathbb{R}$ 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 $J_{x,y}\asymp |x-y|^{-δ}$, 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. |
| title | Phase transitions for contact processes on one-dimensional networks |
| topic | Probability 60K35 (primary), 05C82, 91D30 (secondary) |
| url | https://arxiv.org/abs/2501.16858 |