On the distribution of the critical values of random spherical harmonics

TL;DR

This paper investigates the distribution of critical points and extrema of high-energy random spherical harmonics, deriving their density functions, variances, and demonstrating convergence of empirical measures to expected values.

Contribution

It provides new analytic expressions for the densities and variances of critical points and extrema, and addresses the validity of the Kac-Rice formula in degenerate cases.

Findings

Density functions of extrema and saddles derived

Variances of critical points computed

Empirical measures converge to expected values

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.