Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³

Stecconi, Michele

doi:10.1090/tpms/1177

Isotropic random spin weighted functions on $S^2$ vs isotropic random fields on $S^3$

Author: Michele Stecconi
Journal: Theor. Probability and Math. Statist. 107 (2022), 77-109
MSC (2020): Primary 20C35, 60G60; Secondary 33C55, 53C20, 60B20
DOI: https://doi.org/10.1090/tpms/1177
Published electronically: November 8, 2022
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Abstract:

We show that an isotropic random field on $SU(2)$ is not necessarily isotropic as a random field on $S^3$, although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on $S^3$ is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree $d$ is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range $\bigl \{-\frac {d}{2},\dots ,\frac {d}{2}\bigr \}$, each of which is isotropic in the sense of $SU(2)$. Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.

In addition we will give an overview of the theory of spin weighted functions and Wigner $D$-matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators $\eth \overline {\eth }$ and the horizontal Laplacian of the Hopf fibration $S^3\to S^2$, in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]

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