We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré 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–Poincaré 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.
"A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions." Ann. Probab. 46 (6) 3188 - 3228, November 2018. https://doi.org/10.1214/17-AOP1245