Mean-field behaviour of the random connection model on hyperbolic space
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912641753022464 |
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| author | Dickson, Matthew Heydenreich, Markus |
| author_facet | Dickson, Matthew Heydenreich, Markus |
| contents | We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $λ>0$. Upon variation of $λ$ there is a percolation phase transition: there exists a critical value $λ_c>0$ such that for $λ<λ_c$ all clusters are finite, but infinite clusters exist for $λ>λ_c$. We identify certain critical exponents that characterize the clusters at (and near) $λ_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean-field behaviour of the random connection model on hyperbolic space Dickson, Matthew Heydenreich, Markus Probability 60K35, 82B43, 60G55 We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $λ>0$. Upon variation of $λ$ there is a percolation phase transition: there exists a critical value $λ_c>0$ such that for $λ<λ_c$ all clusters are finite, but infinite clusters exist for $λ>λ_c$. We identify certain critical exponents that characterize the clusters at (and near) $λ_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram. |
| title | Mean-field behaviour of the random connection model on hyperbolic space |
| topic | Probability 60K35, 82B43, 60G55 |
| url | https://arxiv.org/abs/2505.09025 |