Multijet bundles and application to the finiteness of moments for zeros of Gaussian fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909644164694016 |
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| author | Ancona, Michele Letendre, Thomas |
| author_facet | Ancona, Michele Letendre, Thomas |
| contents | We define a notion of multijet for functions on $\mathbb{R}^n$, which extends the classical notion of jets in the sense that the multijet of a function is defined by contact conditions at several points. For all $p \geq 1$ we build a vector bundle of $p$-multijets, defined over a well-chosen compactification of the configuration space of $p$ distinct points in $\mathbb{R}^n$. As an application, we prove that the linear statistics associated with the zero set of a centered Gaussian field on a Riemannian manifold have a finite $p$-th moment as soon as the field is of class~$\mathcal{C}^p$ and its $(p-1)$-jet is nowhere degenerate. We prove a similar result for the linear statistics associated with the critical points of a Gaussian field and those associated with the vanishing locus of a holomorphic Gaussian field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10659 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multijet bundles and application to the finiteness of moments for zeros of Gaussian fields Ancona, Michele Letendre, Thomas Differential Geometry Metric Geometry Probability We define a notion of multijet for functions on $\mathbb{R}^n$, which extends the classical notion of jets in the sense that the multijet of a function is defined by contact conditions at several points. For all $p \geq 1$ we build a vector bundle of $p$-multijets, defined over a well-chosen compactification of the configuration space of $p$ distinct points in $\mathbb{R}^n$. As an application, we prove that the linear statistics associated with the zero set of a centered Gaussian field on a Riemannian manifold have a finite $p$-th moment as soon as the field is of class~$\mathcal{C}^p$ and its $(p-1)$-jet is nowhere degenerate. We prove a similar result for the linear statistics associated with the critical points of a Gaussian field and those associated with the vanishing locus of a holomorphic Gaussian field. |
| title | Multijet bundles and application to the finiteness of moments for zeros of Gaussian fields |
| topic | Differential Geometry Metric Geometry Probability |
| url | https://arxiv.org/abs/2307.10659 |