Normal to Poisson phase transition for subgraph counting in the random-connection model
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917068102696960 |
|---|---|
| author | Liu, Qingwei Privault, Nicolas |
| author_facet | Liu, Qingwei Privault, Nicolas |
| contents | We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $λ$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $λ^{-α}$ we identify a phase transition phenomenon depending on a critical decay rate $α^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $α\in (0,α^\ast_m (G) )$, and a Poisson limit result holds if $α= α^\ast_m (G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal to Poisson phase transition for subgraph counting in the random-connection model Liu, Qingwei Privault, Nicolas Probability 60F05, 60D05, 05C80, 60G55 We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $λ$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $λ^{-α}$ we identify a phase transition phenomenon depending on a critical decay rate $α^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $α\in (0,α^\ast_m (G) )$, and a Poisson limit result holds if $α= α^\ast_m (G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts. |
| title | Normal to Poisson phase transition for subgraph counting in the random-connection model |
| topic | Probability 60F05, 60D05, 05C80, 60G55 |
| url | https://arxiv.org/abs/2409.16222 |