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

TL;DR

This paper introduces multijet bundles to analyze the zeros of Gaussian fields, establishing conditions under which their linear statistics have finite moments, with applications to Gaussian fields on manifolds and holomorphic fields.

Contribution

It defines multijet bundles extending classical jets, and applies this to prove finiteness of moments for linear statistics of Gaussian field zeros and critical points.

Findings

Finite p-th moments for zero sets of Gaussian fields with sufficient regularity.

Extension of jet theory to multijets for multiple contact conditions.

Application to holomorphic Gaussian fields and critical points analysis.

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.