Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics

Cammarota, V.; Marinucci, D.; Wigman, I.

doi:10.1090/proc/13299

Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics
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Proc. Amer. Math. Soc. 144 (2016), 4759-4775 Request permission

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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