Abstract
We study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random L^2-norm of the eigenfunctions is asymptotically one.
Highlights
It can be instructive to compare the results in this paper with other recent characterizations which have been given for the asymptotic distribution for the nodal length of random eigenfunctions in the non-spherical case
We recall first that a non-central limit theorem for arithmetic random waves, i.e. Gaussian Laplacian eigenfunctions on the standard two-dimensional flat torus T2 := R2/Z2, was established in [18]
Analogously to our discussion above the nodal length was decomposed into chaotic components
Summary
[19] provided a stronger characterization of the nodal length fluctuation around their expected value They were able to establish the asymptotic equivalence (in the L2(Ω, F , P) =: L2(P)-sense) of the nodal length and the so-called sample trispectrum of f , i.e. integral over the sphere of H4(f ) that is the fourth-order Hermite polynomial evaluated at the field itself (we recall that H4(t) = t4 − 6t2 + 3). A natural further question to investigate is how much the nodal length behaviour characterizes the behaviour of the boundary length and more generally the full geometry of eigenfunctions, i.e. the behaviour of Lipschitz–Killing curvatures of excursion sets for arbitrary levels u ∈ R. Note that (ii) in Remarks 1.2 and (1.11) immediately give a CLT for the normalized boundary length
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