Critical Points of Chi-Fields

Marinucci, Domenico; STECCONI, Michele

doi:10.30757/ALEA.v22-29

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Critical Points of Chi-Fields

Marinucci, Domenico; STECCONI, Michele

2025 • In Alea, 22 (1), p. 749 - 773

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https://hdl.handle.net/10993/66875

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10.30757/ALEA.v22-29

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The research leading to this paper has been supported by PRIN project Grafia (CUP: E53D23005530006), PRIN Department of Excellence MatMod@Tov (CUP: E83C23000330006) and by the Luxembourg National Research Fund (Grant: 021/16236290/HDS). DM is a member of Indam/GNAMPA.

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Keywords :

33C55; 53C65; 58K05; 60G60; Chi-square; Critical Points; Kac-Rice Formula; Random Fields; Statistics and Probability

Abstract :

[en] We give here a semi-analytic formula for the density of critical values for chi random fields on a general manifold. The result uses Kac-Rice argument and a convenient representation for the Hessian matrix of chi fields, which makes the computation of their expected determinant much more feasible. In the high-threshold limit, the expression for the expected value of critical points becomes very transparent: up to explicit constants, it amounts to Hermite polynomials times a Gaussian density. Our results are also motivated by the analysis of polarization random fields in Cosmology, but they might lead to applications in many different environments.

Disciplines :

Mathematics

Author, co-author :

Marinucci, Domenico; University of Rome Tor vergata, Italy

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

External co-authors :

yes

Language :

English

Title :

Critical Points of Chi-Fields

Publication date :

2025

Journal title :

Alea

ISSN :

1517-106X

Publisher :

Instituto Nacional de Matematica Pura e Aplicada

Volume :

Issue :

Pages :

749 - 773

Peer reviewed :

Peer Reviewed verified by ORBi

Funding text :

Received by the editors September 30th, 2024; accepted April 28th, 2025. 1991 Mathematics Subject Classification. 60G60, 33C55, 53C65, 58K05 . Key words and phrases. Random Fields, Critical Points, Chi-square, Kac-Rice Formula. The research leading to this paper has been supported by PRIN project Grafia (CUP: E53D23005530006), PRIN Department of Excellence MatMod@Tov (CUP: E83C23000330006) and by the Luxembourg National Research Fund (Grant: 021/16236290/HDS). DM is a member of Indam/GNAMPA..

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