A QUANTITATIVE CENTRAL LIMIT THEOREM FOR THE EULER-POINCARE CHARACTERISTIC OF RANDOM SPHERICAL EIGENFUNCTIONS

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler-Poincare characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincare characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, that is, the Euler-Poincare characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz-Killing curvatures of the excursion sets for Gaussian spherical harmonics.