Three central limit theorems for the unbounded excursion component of a Gaussian field
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912916410728448 |
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| author | McAuley, Michael |
| author_facet | McAuley, Michael |
| contents | For a smooth, stationary Gaussian field $f$ on Euclidean space with fast correlation decay, there is a critical level $\ell_c$ such that the excursion set $\{f\geq\ell\}$ contains a (unique) unbounded component if and only if $\ell<\ell_c$. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e. all $\ell<\ell_c$). In higher dimensions the results hold at all sufficiently low levels (all $\ell<-\ell_c<\ell_c$) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three central limit theorems for the unbounded excursion component of a Gaussian field McAuley, Michael Probability 60G60, 60G15, 58K05 For a smooth, stationary Gaussian field $f$ on Euclidean space with fast correlation decay, there is a critical level $\ell_c$ such that the excursion set $\{f\geq\ell\}$ contains a (unique) unbounded component if and only if $\ell<\ell_c$. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e. all $\ell<\ell_c$). In higher dimensions the results hold at all sufficiently low levels (all $\ell<-\ell_c<\ell_c$) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem. |
| title | Three central limit theorems for the unbounded excursion component of a Gaussian field |
| topic | Probability 60G60, 60G15, 58K05 |
| url | https://arxiv.org/abs/2403.03033 |