  We prove a Central Limit Theorem for the Critical Points of Random Spherical
Harmonics, in the High-Energy Limit. The result is a consequence of a deeper
characterizations of the total number of critical points, which are shown to be
asymptotically fully correlated with the sample trispectrum, i.e., the integral
of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As
a consequence, the total number of critical points and the nodal length are
fully correlated for random spherical harmonics, in the high-energy limit.
