Density of critical points for a Gaussian random function

TL;DR

This paper investigates the distribution of critical points in Gaussian random functions, showing that in 2D they are equally distributed between saddle points and extrema, while in 3D saddle points are more frequent.

Contribution

It extends the understanding of critical point densities from 2D to 3D Gaussian fields, revealing a shift in distribution ratios.

Findings

In 2D, saddle points and extrema are equally frequent.

In 3D, saddle points are more common than extrema.

The distribution ratio varies with dimension, breaking topological symmetry.

Abstract

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words, the density of the saddle points and extrema agree. This argument breaks down in three dimensions. All ratios of the densities of saddle points and extrema larger than one are possible. For a homogeneous Gaussian random field one finds no longer an equidistribution of signs, saddle points are slightly more frequent.