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Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres

Abstract: We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere Sd. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on Sd. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated with isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble (in S2). © 2016 Elsevier Inc.

 Fuente: Journal of Complexity 37 (2016) 76-109

 Editorial: Academic Press Inc.

 Año de publicación: 2016

 Nº de páginas: 34

 Tipo de publicación: Artículo de Revista

 DOI: 10.1016/j.jco.2016.08.001

 ISSN: 0885-064X,1090-2708

 Proyecto español: MTM2014-51834-P ; MTM2014-57590-P

 Url de la publicación: http://dx.doi.org/10.1016/j.jco.2016.08.001