Theory of Probability and Mathematical Statistics
Isotropic random spin weighted functions on S2 vs isotropic random fields on S3
Michele Stecconi
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Abstract: We show that an isotropic random field on SU(2) is not necessarily isotropic as a random field on S3, 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 S3 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 {-d/2,...,d/2}, 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 ðð and the horizontal Laplacian of the Hopf fibration S3→S2, in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]
Keywords: Isotropic random fields, Riemannian geometry, spherical harmonics, random waves, Wigner D-matrices
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