Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909829379915776 |
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| author | Dickson, Matthew |
| author_facet | Dickson, Matthew |
| contents | A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models on the $d$-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic random connection models. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some random connection models whose resulting graphs are almost surely not locally finite. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform Dickson, Matthew Probability 82B43 (Primary) 60G55, 43A90 (Secondary) A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models on the $d$-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic random connection models. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some random connection models whose resulting graphs are almost surely not locally finite. |
| title | Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform |
| topic | Probability 82B43 (Primary) 60G55, 43A90 (Secondary) |
| url | https://arxiv.org/abs/2412.12854 |