Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics (Q2821738)

The Gaussian kinematic formula (GKF) yields a precise expression for the expected value of the Euler-Poincare characteristic (EPC) of excursion sets of smooth Gaussian processes. Different methods have been to estimate numerically the covariance matrix of the EPC characteristic for the joint excursion sets at various thresholds. In this paper the authors establish analytic formulae for the covariance of the EPC characteristic of excursion sets at different thresholds focusing on Gaussian spherical harmonics. The established expression seems to be related to a second-order Gaussian kinematic formula. The proof of their main theorem follows from Morse theory and the analysis of asymptotic fluctuations of critical points of random eigenfunctions.