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Peer-reviewedCC BYAnnales Henri Lebesgue

The Triangle Condition for the Marked Random Connection Model

  • Matthew Dickson1
  • Markus Heydenreich2
  1. 1University College Dublin, School of Mathematics and Statistics, Belfield, Dublin 4 (Ireland)
  2. 2Universität Augsburg, Institut für Mathematik, 86135 Augsburg (Germany)
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Abstract

We investigate a spatial random graph model whose vertices are given as a marked Poisson process on ℝ d . Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely. Our focus is on the transition between the low- and high-density phase, where the system is critical. We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, and thus mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.

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Cite

BibTeXRIS
BibTeX
@article{dickson2026triangle,
  title = {{The Triangle Condition for the Marked Random Connection Model}},
  author = {Matthew Dickson and Markus Heydenreich},
  journal = {Annales Henri Lebesgue},
  year = {2026},
  month = oct,
  doi = {10.5802/ahl.285},
  url = {https://doi.org/10.5802/ahl.285},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The Triangle Condition for the Marked Random Connection Model
AU  - Matthew Dickson
AU  - Markus Heydenreich
PY  - 2026
DA  - 2026/10/21
JO  - Annales Henri Lebesgue
DO  - 10.5802/ahl.285
UR  - https://doi.org/10.5802/ahl.285
AB  - We investigate a spatial random graph model whose vertices are given as a marked Poisson process on ℝ d . Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely. Our focus is on the transition between the low- and high-density phase, where the system is critical. We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, and thus mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.
SN  - 2644-9463
ER  -