Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918356790018048 |
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| author | Gracar, Peter Jahnel, Benedikt Lüchtrath, Lukas Vu, Anh Duc |
| author_facet | Gracar, Peter Jahnel, Benedikt Lüchtrath, Lukas Vu, Anh Duc |
| contents | We consider a dynamic network in continuum time and space in which nodes, with initial locations given by a Poisson point process, move according to i.i.d. isotropic $α$-stable processes. Each node is additionally equipped with an i.i.d. detection radius. Inspired by corresponding results by Peres et. al. on mobile networks based on Brownian sausages with fixed width, we investigate the tail behaviour of three stopping times: The detection time of the first discovery of a designated node, the first coverage of an entire set, and the first discovery of a node by the infinite connected component of the system. Broadly speaking, we discover that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates. The proofs rest on heat-kernel bounds for the underlying Lévy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22109 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes Gracar, Peter Jahnel, Benedikt Lüchtrath, Lukas Vu, Anh Duc Probability 60D05, secondary: 60K35 We consider a dynamic network in continuum time and space in which nodes, with initial locations given by a Poisson point process, move according to i.i.d. isotropic $α$-stable processes. Each node is additionally equipped with an i.i.d. detection radius. Inspired by corresponding results by Peres et. al. on mobile networks based on Brownian sausages with fixed width, we investigate the tail behaviour of three stopping times: The detection time of the first discovery of a designated node, the first coverage of an entire set, and the first discovery of a node by the infinite connected component of the system. Broadly speaking, we discover that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates. The proofs rest on heat-kernel bounds for the underlying Lévy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system. |
| title | Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes |
| topic | Probability 60D05, secondary: 60K35 |
| url | https://arxiv.org/abs/2602.22109 |