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Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics


Authors: V. Cammarota, D. Marinucci and I. Wigman
Journal: Proc. Amer. Math. Soc. 144 (2016), 4759-4775
MSC (2010): Primary 33C55, 42C10, 60D05, 60B10, 60G60
DOI: https://doi.org/10.1090/proc/13299
Published electronically: August 1, 2016
MathSciNet review: 3544528
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Abstract: In this short note, we build upon recent results from our earlier paper to present a precise expression for the asymptotic variance of the Euler-Poincaré characteristic for the excursion sets of Gaussian eigenfunctions on $ \mathcal {S}^2$; this result can be written as a second-order Gaussian kinematic formula for the excursion sets of random spherical harmonics. The covariance between the Euler-Poincaré characteristics for different level sets is shown to be fully degenerate; it is also proved that the variance for the zero level excursion sets is asymptotically of smaller order.


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Additional Information

V. Cammarota
Affiliation: Department of Mathematics, Università degli Studi di Roma Tor Vergata, 00133 Rome, Italy
Email: cammarot@mat.uniroma2.it

D. Marinucci
Affiliation: Department of Mathematics, Università degli Studi di Roma Tor Vergata, 00133 Rome, Italy
Email: marinucc@mat.uniroma2.it

I. Wigman
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2 2LS, England
Email: igor.wigman@kcl.ac.uk

DOI: https://doi.org/10.1090/proc/13299
Received by editor(s): April 8, 2015
Received by editor(s) in revised form: December 22, 2015
Published electronically: August 1, 2016
Additional Notes: The first and second author’s research was supported by ERC grant No. 277742
The third author’s research was supported by ERC grant No. 335141
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society