Fluctuations of the Euler-Poincare` Characteristic for Random Spherical Harmonics

In this short note, we build upon recent results from [7] to present a precise expression for the asymptotic variance of the Euler-Poincar´e characteristic for the excursion sets of Gaussian eigenfunctions on 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-Poincar´e 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