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We prove a Central Limit Theorem for the Critical Points of Random SphericalHarmonics, in the High-Energy Limit. The result is a consequence of a deepercharacterizations of the total number of critical points, which are shown to beasymptotically fully correlated with the sample trispectrum, i.e., the integralof the fourth Hermite polynomial evaluated on the eigenfunctions themselves. Asa consequence, the total number of critical points and the nodal length arefully correlated for random spherical harmonics, in the high-energy limit.

On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics