The number of critical points of a Gaussian field: finiteness of moments

46B70; 58K05; 60D05; 60F05; 60F25; 60G15; Critical points; Gaussian field; Moments; Nodal volume; Analysis; Statistics and Probability; Statistics, Probability and Uncertainty

Abstract :

[en] Let f be a Gaussian random field on Rd and let X be the number of critical points of f contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on f, the random variable X has finite moments. So far, this has been established only for moments of order lower than three. In this paper, we prove the conjecture. Precisely, we show that X has finite moment of order p, as soon as, at any given point, the Taylor polynomial of order p of f is non-degenerate. We present a simple and general approach that is not specific to critical points and we provide various applications. In particular, we show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble.

Disciplines :

Mathematics

Author, co-author :

GASS, Louis ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

STECCONI, Michele ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

Language :

English

Title :

The number of critical points of a Gaussian field: finiteness of moments

Publication date :

December 2024

Journal title :

Probability Theory and Related Fields

ISSN :

0178-8051

eISSN :

1432-2064

Publisher :

Springer Science and Business Media Deutschland GmbH

Volume :

190

Issue :

3-4

Pages :

1167 - 1197

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.springer.com/content/pdf/10.1007/s00440-024-01273-5.pdf

Funders :

Luxembourg National Research Fund

Funding text :

This work was supported by the Luxembourg National Research Fund (Grant: 021/16236290/HDSA).

Available on ORBilu :

since 16 December 2025

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