On the Distribution of the Critical Values of Random Spherical Harmonics

Abstract

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.