Betti numbers; Lipschitz–Killing curvatures; Spin random eigenfunctions; Spin random fields; Analysis; Algebra and Number Theory; Mathematical Physics
Abstract :
[en] 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.
Agence Nationale de la Recherche Conseil Régional des Pays de la Loire Fonds National de la Recherche Luxembourg Dipartimenti di Eccellenza Istituto Nazionale di Alta Matematica "Francesco Severi"
Funding text :
The authors would like to thank an anonymous referee for insightful remarks and useful suggestions. DM acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome \u201CTor Vergata\u201D, CUP E83C18000100006 and INdAM. The research of MR has been supported by the ANR-17-CE40-0008 Project UNIRANDOM and INdAM. MS is supported by the grant TROPICOUNT of R\u00E9gion Pays de la Loire, the ANR Project ENUMGEOM NR-18-CE40-0009-02 and by the Luxembourg National Research Fund (Grant 021/16236290/HDSA).
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