Critical Exponents for Marked Random Connection Models
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915383483564032 |
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| author | Caicedo, Alejandro Dickson, Matthew |
| author_facet | Caicedo, Alejandro Dickson, Matthew |
| contents | Here we prove critical exponents for Random Connections Models (RCMs) with random marks. The vertices are given by a marked Poisson point process on $\mathbb{R}^d$ and an edge exists between any pair of vertices independently with a probability depending upon their spatial displacement and on their respective marks. Given conditions on the edge probabilities, we prove mean-field lower bounds for the susceptibility and percolation functions. In particular, we prove the equality of the susceptibility and percolation critical intensities. If we assume that a form of the triangle condition holds, then we also prove that the susceptibility, percolation and cluster tail critical exponents exist and take their mean-field values. Our proof approach adapts the differential inequality and magnetization function approaches that have been previously applied to discrete homogeneous settings to our continuum marked setting. This includes a proof of the analyticity of the magnetization function in the required parameter regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07398 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Critical Exponents for Marked Random Connection Models Caicedo, Alejandro Dickson, Matthew Probability 60K35, 82B43 (Primary) 60G55 (Secondary) Here we prove critical exponents for Random Connections Models (RCMs) with random marks. The vertices are given by a marked Poisson point process on $\mathbb{R}^d$ and an edge exists between any pair of vertices independently with a probability depending upon their spatial displacement and on their respective marks. Given conditions on the edge probabilities, we prove mean-field lower bounds for the susceptibility and percolation functions. In particular, we prove the equality of the susceptibility and percolation critical intensities. If we assume that a form of the triangle condition holds, then we also prove that the susceptibility, percolation and cluster tail critical exponents exist and take their mean-field values. Our proof approach adapts the differential inequality and magnetization function approaches that have been previously applied to discrete homogeneous settings to our continuum marked setting. This includes a proof of the analyticity of the magnetization function in the required parameter regime. |
| title | Critical Exponents for Marked Random Connection Models |
| topic | Probability 60K35, 82B43 (Primary) 60G55 (Secondary) |
| url | https://arxiv.org/abs/2305.07398 |