Geometry and topology of spin random fields

Abstract

Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be viewed as random sections of the s-th complex tensor power of the tangent bundle of the 2-sphere. In this paper, we discuss how to characterize their expected geometry and topology. In particular, we investigate the asymptotic behaviour, under scaling assumptions, of general classes of geometric and topological functionals including Lipschitz–Killing Curvatures and Betti numbers for (properly defined) excursion sets; we cover both the cases of fixed and diverging spin parameters s. In the special case of monochromatic fields (i.e., spin random eigenfunctions) our results are particularly explicit; we show how their asymptotic behaviour is non-universal and we can obtain in particular complex versions of Berry’s random waves and of Bargmann–Fock’s models as subcases of a new generalized model, depending on the rate of divergence of the spin parameter s.