Critical Points of Chi-Fields

TL;DR

This paper derives a semi-analytic formula for the density of critical values of chi random fields on manifolds, simplifying calculations and revealing explicit high-threshold behavior with potential applications in cosmology.

Contribution

It introduces a new semi-analytic approach using Kac-Rice and a Hessian representation to compute critical value densities for chi fields on manifolds.

Findings

Explicit formula for critical value density using Kac-Rice

Simplified high-threshold limit involving Hermite polynomials

Potential applications in cosmological polarization fields

Abstract

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.