[Paper Review] Geometry and topology of spin random fields

Antonio Lerario, Domenico Marinucci+2 more|arXiv (Cornell University)|Jul 18, 2022

Historical Geography and Cartography4 citations

TL;DR

This paper develops a general framework to characterize the expected geometry and topology of spin random fields—critical in cosmology for modeling Cosmic Microwave Background and polarization data—by analyzing asymptotic behaviors of Lipschitz-Killing curvatures and Betti numbers of excursion sets under scaling limits. It reveals non-universal asymptotic geometry depending on the spin parameter's growth rate, recovering complex random waves and Bargmann-Fock models as limiting cases.

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.

Motivation & Objective

• To develop a general method for characterizing the expected geometry and topology of spin random fields on the 2-sphere.
• To study the asymptotic behavior of geometric and topological functionals—such as Lipschitz-Killing curvatures and Betti numbers—for excursion sets of spin random fields.
• To investigate how the asymptotic geometry depends on the scaling of the spin parameter $ s $, particularly in the monochromatic (eigenfunction) case.
• To unify and generalize existing models such as Berry’s complex random waves and Bargmann-Fock fields as limiting cases of a broader class of spin random eigenfunctions.

Proposed method

• The authors model spin random fields as random sections of the $ s $-th tensor power of the tangent bundle on $ \mathbb{S}^2 $, using a spectral representation via spherical harmonics.
• They define excursion sets and apply stratified Morse theory to analyze topological invariants like Betti numbers and Lipschitz-Killing curvatures.
• A scaling assumption is introduced on the covariance function, leading to a rescaled field whose convergence is analyzed in distribution and in expectation.
• The analysis relies on intrinsic differential geometry, including jet bundles and Whitney stratified subsets, to characterize type-$ W $ singularities and their contributions to curvature and Betti number functionals.
• The authors derive explicit formulas for the first few intrinsic volumes (e.g., Euler-Poincaré characteristic, area) via semialgebraic Morse inequalities and integration over singular sets.
• Technical tools include Hilb-type asymptotics for Bessel functions and convergence results for the covariance function under different regimes of $ s $.

Experimental results

Research questions

• RQ1 How do the expected Lipschitz-Killing curvatures of excursion sets of spin random fields behave asymptotically as the spin parameter $ s $ grows?
• RQ2 What is the asymptotic behavior of Betti numbers of excursion sets in the limit of large spin, and how does it depend on the rate of divergence of $ s $?
• RQ3 Can complex versions of Berry’s random waves and Bargmann-Fock models be recovered as limiting cases of a generalized spin random eigenfunction model?
• RQ4 What is the role of the spin parameter’s growth rate in determining the universality or non-universality of the asymptotic geometric and topological behavior?
• RQ5 How do the topological invariants of the zero set and critical points of spin random eigenfunctions scale with $ s $, and what are their limiting distributions?

Key findings

• The asymptotic behavior of geometric and topological functionals is non-universal and depends on the rate at which the spin parameter $ s $ diverges.
• When $ s $ grows slowly (fixed $ r $ in the scaling regime), the limiting distribution corresponds to a generalized model $ M_r(x) $, which includes the real part of the complex Bargmann-Fock field for $ r=0 $.
• When $ s $ grows rapidly (i.e., $ r \to \infty $), the limiting distribution converges to $ J_0(x) $, corresponding to the complex version of Berry’s random wave model.
• The expected Euler-Poincaré characteristic and first intrinsic volume of excursion sets converge to explicit limits under the scaling assumptions, with the area (second intrinsic volume) also converging to a deterministic limit.
• For monochromatic spin eigenfunctions, the Betti numbers of excursion sets converge in expectation to a limit that depends on the scaling of $ s $, with different asymptotic regimes yielding distinct topological behaviors.
• The results generalize classical random wave models and provide a unified framework for analyzing topological and geometric functionals in cosmological data analysis.