Crossings and diffusion in Poisson driven marked random connection models
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912673282654208 |
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| author | Faggionato, Alessandra Hartarsky, Ivailo |
| author_facet | Faggionato, Alessandra Hartarsky, Ivailo |
| contents | We first study crossing statistics in random connection models (RCM) built on marked Poisson point processes on $\mathbb R^d$. Under general assumptions, we show exponential tail bounds for the number of crossings of a box contained in the infinite cluster for supercritical intensity of the point process, and percolation in slabs, in analogy with the Grimmett-Marstrand theorem. We then present several applications to transport and diffusion phenomena. In particular, we prove the non-degeneracy of the effective homogenized matrix arising in the large-scale limit of random walks, exclusion processes, and resistor networks on the RCM, and the non-degeneracy of the effective diffusion constant for one-dimensional diffusion operators on the Euclidean graph associated with the RCM. As examples, we apply our results to Poisson-Boolean models and Mott variable range hopping random resistor network, providing a fundamental ingredient used in the derivation of Mott's law. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_03965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crossings and diffusion in Poisson driven marked random connection models Faggionato, Alessandra Hartarsky, Ivailo Probability 60K35, 60K37, 60D05, 82B21 We first study crossing statistics in random connection models (RCM) built on marked Poisson point processes on $\mathbb R^d$. Under general assumptions, we show exponential tail bounds for the number of crossings of a box contained in the infinite cluster for supercritical intensity of the point process, and percolation in slabs, in analogy with the Grimmett-Marstrand theorem. We then present several applications to transport and diffusion phenomena. In particular, we prove the non-degeneracy of the effective homogenized matrix arising in the large-scale limit of random walks, exclusion processes, and resistor networks on the RCM, and the non-degeneracy of the effective diffusion constant for one-dimensional diffusion operators on the Euclidean graph associated with the RCM. As examples, we apply our results to Poisson-Boolean models and Mott variable range hopping random resistor network, providing a fundamental ingredient used in the derivation of Mott's law. |
| title | Crossings and diffusion in Poisson driven marked random connection models |
| topic | Probability 60K35, 60K37, 60D05, 82B21 |
| url | https://arxiv.org/abs/2507.03965 |