Multijet bundles and application to the finiteness of moments for zeros of Gaussian fields

Author:

Thomas Letendre, Michele Ancona

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Metric Geometry (math.MG), Probability (math.PR)

journal:

date:

2023-07-19 16:00:00

Abstract

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.

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