Gracar, P. orcid.org/0000-0001-8340-8340, Lüchtrath, L. and Mönch, C. (2025) Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension. Electronic Journal of Probability, 30. pp. 1-29. ISSN: 1083-6489
Abstract
We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient δeff which quantifies the influence of heavy-tailed degrees on long-range connections. We show that δeff < 2 is sufficient for the existence of a supercritical percolation phase in the model and that δeff > 2 always implies the absence of percolation. In particular, our results complement those in Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given for the soft Boolean model and the age-dependent random connection model for both the existence and the absence of a subcritical percolation phase. Our results further provide a criterion for the existence or non-existence of a giant component in large finite graphs.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2024 Project Euclid. This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | weight-dependent random connection model; spatial random graphs; scale-free degree distribution; soft Boolean model; age-dependent random connection model; scale-free percolation; phase transition; giant component |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 24 Sep 2025 10:52 |
Last Modified: | 24 Sep 2025 10:52 |
Status: | Published |
Publisher: | Institute of Mathematical Statistics |
Identification Number: | 10.1214/25-EJP1399 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:231953 |