On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics
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Annales Henri Poincar´ e
On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics Domenico Marinucci
and Maurizia Rossi
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 L2 -norm of the eigenfunctions is asymptotically one. Mathematics Subject Classification. 60G60, 62M15, 53C65, 42C10, 33C55.
1. Introduction and Main Result 1.1. Random Spherical Harmonics On the unit two-dimensional sphere S2 , let us consider the Helmholtz equation f : S2 → R,
ΔS2 f + λ f = 0 , where ΔS2
1 ∂ = sin θ ∂θ
∂ sin θ ∂θ
+
1 ∂2 sin2 θ ∂ϕ2
is the Laplace–Beltrami operator on S2 in spherical coordinates (θ, ϕ) and {λ = ( + 1)}∈N represent the set of eigenvalues of −ΔS2 . For any λ , the corresponding eigenspace is the (2 + 1)-dimensional space of spherical harmonics of degree ; we can choose the standard L2 -orthonormal basis made of the spherical harmonics {Ym }m=−,..., [13, §3.4] and focus, for ∈ N∗ , on random eigenfunctions of the form 4π a,m Ym (x) , x ∈ S2 . (1.1) f (x) = 2 + 1 m=−
D. Marinucci, M. Rossi
Ann. Henri Poincar´e
Here the coefficients {a,m }m=−,..., are random variables (defined on some probability space (Ω, F, P) that we fix once and for all) such that a,0 is a real standard Gaussian, and for m = 0 the a,m ’s are standard complex Gaussians (independent of a,0 ) and independent save for the relation a,m = (−1)m a,−m ensuring f to be real valued—it is immediate to see that the law of the process f is invariant with respect to the choice of a L2 -orthonormal basis of eigenfunctions. For every , the random eigenfunction f is centred, Gaussian and isotropic; from the addition theorem for spherical harmonics [13, (3.42)], the covariance function is given by r (x, y) := E[f (x)f (y)] = P (cos d(x, y)),
x, y ∈ S2 ,
(1.2)
where P is the th Legendre polynomial, defined by Rodrigues’ formula, as 1 d 2 (t − 1) , t ∈ [−1, 1], 2 ! dt and d(x, y) stands for the spherical geodesic distance between the points x and y. As discussed elsewhere (see i.e. [3,4,7]), random spherical harmonics arise naturally from the spectral analysis of isotropic Gaussian fields on the sphere or in the investigation of quantum chaos (see for instance [13,29] for reviews). P (t) :=
Remark 1.1. The random variables {a,m }m=−,...,,∈N∗ are defined on the same probability space (Ω, F, P). We may assume f and f in (1.1) to be independent random fields whenever = , this is equivalent to assume that a,m and a ,m are independent random variables for every m, m whenever = . In this paper, we shall focus on excursion sets of f in (1.1) defined as Au (f ) := {x ∈ S2 : f (x) ≥ u} for u ∈ R. The boundary of Au (f ), i.e. the level set f−1 (u) := {x ∈ S2 : f (x) = u}, is an a.s. smooth curve whose connected components are homeomorphic to the circumferenc
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