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.

A reduction principle for the critical values of random spherical harmonics

“…The following result was given in [7]: for every interval I ⊆ R as ℓ → ∞ Var(N c ℓ (I)) = ℓ 3 ν c (I) + O(ℓ 5/2 ) , again with a universal error bound in the O(·) term; similar results hold for the number of extrema and saddles. Remark 1.1.…”

A second moment bound for critical points of planar Gaussian fields in shrinking height windows

“…A simple application of the Kac-Rice formula shows that, under mild conditions on κ, the mean of N R [a, b] is of order O(R 2 (b − a)), and in fact it is not difficult to compute asymptotics for E[N R [a, b]]/R 2 explicitly (see, e.g., [6,8] for special cases). On the other hand, the second moment of N R [a, b] is a more difficult quantity to control, and indeed its finiteness was only established recently [10] (see also [9,11]), with the finiteness of higher moments remaining an important open question.…”

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

“…Of course, in order to exploit Lipschitz-Killing curvatures/Minkowski functionals to implement data analysis tools the expected value by itself is not sufficient, but we need also analytic expression for the variance. The latter was derived in some recent results by Cammarota and Marinucci (2018a) and Cammarota et al (2016b); and see Todino (2018a) for a review.…”

“…As a further tool of investigation, we shall consider in this paper also the behavior of critical points for random spherical harmonics, which has recently been fully characterized by Cammarota and Marinucci (), Cammarota et al (), Cammarota and Wigman (), among others.…”

“…As a further tool of investigation, we shall consider in this paper also the behavior of critical points for random spherical harmonics, which has recently been fully characterized by Cammarota and Marinucci (2018b), Cammarota et al (2016b), Cammarota and Wigman (2017), among others. More precisely, by definition critical points, extrema and saddles are, respectively, given by: where we used a = c, e, s to label critical points, extrema, and saddles respectively.…”