A reduction principle for the critical values of random spherical harmonics

Abstract

We study here the random fluctuations in the number of critical points with values in an interval I ⊂ R for Gaussian spherical eigenfunctions {f ℓ }, in the high energy regime where ℓ → ∞. We show that these fluctuations are asymptotically equivalent to the centred L 2 -norm of {f ℓ } times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.•

Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

“…This procedure will guarantee that the estimator has minimal variance. Corollary 1 is consistent with recent results on the so‐called Berry's cancelation phenomenon, that is, the fact that the variance of the LK curvatures at level s u reaches a minimum, because of the disappearance of a large number of chaotic terms (see Cammarota & Marinucci, 2020).…”

On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

“…Let us stress now that one of the main differences w.r.t. the nodal case (1.6) is that, as argued in [4], thanks to (1.11) the boundary length is asymptotically (as → +∞) perfectly correlated (meaning that the squared correlation coefficients ρ(•, •) converge to one) with other geometric functionals, such as the area [15,16] and the Euler-Poincaré characteristic [4] for the excursion regions A u (f ) = {x ∈ S 2 : f (x) ≥ u}; likewise, perfect correlation holds between these functionals and the number of critical points [5] for f (x) in the same excursion region, for any non-zero threshold u = 0. In other words, at high values of , knowledge of any of these quantities for a given sample yields the full information on the behaviour of all the others for the same sample, up to asymptotically negligible terms.…”

Some Recent Developments on the Geometry of Random Spherical Eigenfunctions

“…Later in [13] it was shown that the critical values above the threshold level satisfy the asymptotic…”