  We study the limiting distribution of critical points and extrema of random
spherical harmonics, in the high energy limit. In particular, we first derive
the density functions of extrema and saddles; we then provide analytic
expressions for the variances and we show that the empirical measures in the
high-energy limits converge weakly to their expected values. Our arguments
require a careful investigation of the validity of the Kac-Rice formula in
nonstandard circumstances, entailing degeneracies of covariance matrices for
first and second derivatives of the processes being analyzed.
