On the number of nodal domains of random spherical harmonics

Abstract

Abstract. Let N (f ) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N (f )/n 2 tends to a positive constant a, and that N (f )/n 2 exponentially concentrates around a. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.

Critical sets of random smooth functions on compact manifolds

“…In this section we also present a simple application of this formula to the number of critical points of random spherical harmonics of large degree on S 2 . This sheds additional light on a recent result of Nazarov and Sodin [30] on the number of nodal domains of random spherical harmonics. More precisely, the inequality (2.40) shows that the expected number δ n of zonal domain on S 2 of a random harmonic polynomial of large degree n satisfies the upper bound δ n < 0.29n 2 .…”

“…Additionally, for large n, with high probability, δ(u) is close to an 2 (see [30] for a precise statement). This shows that…”

On the nodal count statistics for separable systems in any dimension

“…Describing the region W in action-angle variables eventually implies compactness. Using homogeneity one may reduce the s-dimensional integral (22) to an s − 1-dimensional integral over the s − 1-dimensional compact surface (unit energy shell) ∂Γ ≡ {I : H(I) = 1} in momentum space. Note that ∂Γ is the non-trivial part of the boundary of the region Γ and intersects the hyperplanes I l = 0 (which are also boundaries of Γ).…”

Topological Properties of Neumann Domains

“…Let us specify some points of comparison between Neumann domains and nodal domains. From a topological point of view, Neumann domains are simply connected (theorems 1.4(iv) and 3.13(iv)), whereas nodal domains are not in general [33,35]. The simplicity of the Neumann partition is also apparent in the eigenfunction restriction to a Neumann domain, f | Ω .…”