Singularities of Gaussian Random Maps into the Plane

TL;DR

This paper calculates the expected properties of singularities in Gaussian random maps into the plane, providing explicit formulas under various assumptions, which advances understanding of geometric features in random fields.

Contribution

It introduces formulas for expected singularity measures of Gaussian fields on manifolds, including special cases like i.i.d. and isotropic fields, with no prior assumptions beyond smoothness.

Findings

Expected lengths of critical curves and contours

Number of cusps in Gaussian maps

Explicit formulas under specific assumptions

Abstract

We compute the expected value of various quantities related to the biparametric singularities of a pair of smooth centered Gaussian random fields on an n-dimensional compact manifold, such as the lengths of the critical curves and contours of a fixed index and the number of cusps. We obtain certain expressions under no particular assumptions other than smoothness of the two fields, but more explicit formulae are derived under varying levels of additional constraints such as the two random fields being i.i.d, stationary, isotropic etc.