1 Introduction and Main Result

1.1 Random Spherical Harmonics

On the unit two-dimensional sphere \(\mathbb {S}^{2},\) let us consider the Helmholtz equation

$$\begin{aligned} \Delta _{\mathbb {S}^{2}}f_{\ell }+\lambda _{\ell }f_{\ell }=0\text { },f_{\ell }:\mathbb {S}^{2}\rightarrow \mathbb {\mathbb {R}}, \end{aligned}$$

where

$$\begin{aligned} \Delta _{\mathbb {S}^{2}}=\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left( \sin \theta \frac{\partial }{\partial \theta } \right) +\frac{1}{\sin ^{2}\theta }\frac{\partial ^2 }{\partial \varphi ^{2}} \end{aligned}$$

is the Laplace–Beltrami operator on \(\mathbb {S}^{2}\) in spherical coordinates \((\theta ,\varphi )\) and \(\left\{ \lambda _{\ell }=\ell (\ell +1)\right\} _{\ell \in \mathbb N}\) represent the set of eigenvalues of \(-\Delta _{\mathbb {S}^{2}}\). For any \(\lambda _{\ell }\), the corresponding eigenspace is the \((2\ell +1)\)-dimensional space of spherical harmonics of degree \(\ell \); we can choose the standard \(L^{2}\)-orthonormal basis made of the spherical harmonics \(\left\{ Y_{\mathbb {\ell }m}\right\} _{m=-\ell ,\dots , \ell }\) [13, §3.4] and focus, for \(\ell \in \mathbb N^*\), on random eigenfunctions of the form

$$\begin{aligned} f_{\ell }(x)=\sqrt{\frac{4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }a_{\ell , m}Y_{\ell m}(x)\text { ,}\qquad x\in \mathbb S^2. \end{aligned}$$
(1.1)

Here the coefficients \(\left\{ a_{\mathbb {\ell ,}m}\right\} _{m=-\ell ,\dots , \ell }\) are random variables (defined on some probability space \((\Omega , \mathcal F, \mathbb P)\) that we fix once and for all) such that \(a_{\ell ,0}\) is a real standard Gaussian, and for \(m\ne 0\) the \(a_{\ell ,m}\)’s are standard complex Gaussians (independent of \(a_{\ell ,0}\)) and independent save for the relation \(\overline{a_{\ell ,m}} =(-1)^m a_{\ell , -m}\) ensuring \(f_\ell \) to be real valued—it is immediate to see that the law of the process \(f_{\ell }\) is invariant with respect to the choice of a \(L^{2}\)-orthonormal basis of eigenfunctions.

For every \(\ell \), the random eigenfunction \(f_{\ell }\) is centred, Gaussian and isotropic; from the addition theorem for spherical harmonics [13, (3.42)], the covariance function is given by

$$\begin{aligned} r_\ell (x,y):=\mathbb {E}[f_{\ell }(x)f_{\ell }(y)]=P_{\ell }(\cos d(x,y)),\qquad x,y\in \mathbb S^2, \end{aligned}$$
(1.2)

where \(P_{\ell }\) is the \(\ell \)th Legendre polynomial, defined by Rodrigues’ formula, as

$$\begin{aligned} P_{\ell }(t):=\frac{1}{2^{\ell }\ell !}\frac{d^\ell }{dt^{\ell }}(t^{2}-1)^{\ell } \text { , }\qquad t\in [-1, 1], \end{aligned}$$

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).

Remark 1.1

The random variables \(\left\{ a_{\mathbb {\ell ,}m}\right\} _{m=-\ell ,\dots , \ell , \ell \in \mathbb N^*}\) are defined on the same probability space \((\Omega , \mathcal F, \mathbb P)\). We may assume \(f_\ell \) and \(f_{\ell '}\) in (1.1) to be independent random fields whenever \(\ell \ne \ell '\), this is equivalent to assume that \(a_{\ell ,m}\) and \(a_{\ell ',m'}\) are independent random variables for every \(m,m'\) whenever \(\ell \ne \ell '\).

In this paper, we shall focus on excursion sets of \(f_{\ell }\) in (1.1) defined as \( A_u(f_{\ell }):=\{x \in \mathbb {S}^2:f_{\ell }(x) \ge u\}\) for \(u\in \mathbb R\). The boundary of \(A_u(f_\ell )\), i.e. the level set

$$\begin{aligned}f_\ell ^{-1}(u) :=\lbrace x\in \mathbb S^2:f_\ell (x)=u\rbrace ,\end{aligned}$$

is an a.s. smooth curve whose connected components are homeomorphic to the circumference. Let us define, for \(u\in \mathbb {R}\), the random variable

$$\begin{aligned} \mathcal {L}_{\ell }(u):=\text {length}\left( f_{\ell }^{-1}(u)\right) . \end{aligned}$$
(1.3)

If \(u=0\), the quantity \(\mathcal {L}_{\ell }(0)\) is known as nodal length. For \(u\ne 0\), we will often refer to \(\mathcal L_\ell (u)\) in (1.3) as “boundary length”, equivalent with “length of level sets”. In this manuscript, we shall investigate in particular the sequence of random variables \(\lbrace \mathcal {L}_{\ell }(u)\rbrace _{\ell \in \mathbb N^*}\).

1.1.1 Nodal Length

The nodal length \(\{\mathcal L_\ell (0)\}_{\mathcal \ell \in \mathbb N^*}\) has been the object of an enormous amount of activity, see i.e. [2, 19, 26, 28]. In particular, according to the celebrated Yau’s conjecture [31], which has been proved for real analytic manifolds by Donnelly and Fefferman, see [9], and more recently for smooth manifolds by [12] (the lower bound), there exist two constants \(c,C>0\) s.t. for any realization \(f_{\ell }\) one has

$$\begin{aligned} c\sqrt{\lambda _{\ell }}\le \mathcal {L}_{\ell }(0)\le C\sqrt{\lambda _{\ell }} \end{aligned}$$

for every \(\ell \in \mathbb N^*\). For the “typical” eigenfunction, fluctuations are indeed of smaller order; more precisely, tighter bounds can be given in a probabilistic sense. In fact, under Gaussianity the expected value of \(\mathcal {L}_{\ell }(0)\) is easily computed to be (for instance, by the Gaussian Kinematic Formula [1], see also [2])

$$\begin{aligned} \mathbb {E}\left[ \mathcal {L}_{\ell }(0)\right] =2\pi \sqrt{\frac{\lambda _{\ell }}{2}}. \end{aligned}$$

The computation of the variance is much more challenging and was solved by [28], where it is shown that, as \(\ell \rightarrow \infty ,\)

$$\begin{aligned} \mathrm{Var}\left( \mathcal {L}_{\ell }(0)\right) =\frac{\log \ell }{32}+O(1)\text { . } \end{aligned}$$
(1.4)

More recently, [19] actually provided a stronger characterization of the nodal length fluctuation around their expected value. More precisely, they were able to establish the asymptotic equivalence (in the \(L^{2}(\Omega , \mathcal F, \mathbb P ) =: L^{2}(\mathbb P )\)-sense) of the nodal length and the so-called sample trispectrum of \(f_{\ell }\), i.e. integral over the sphere of \(H_{4}(f_{\ell })\) that is the fourth-order Hermite polynomial evaluated at the field itself (we recall that \(H_{4}(t)=t^{4}-6t^{2}+3\)). Indeed, let us define first the sequence of random variables

$$\begin{aligned} \mathcal {M}_{\ell }:=-\frac{1}{4}\sqrt{\frac{\lambda _\ell }{2}}\frac{1}{4!} \int _{\mathbb {S}^{2}}H_{4}(f_{\ell }(x))\,\hbox {d}x,\qquad \ell \in \mathbb N^*; \end{aligned}$$
(1.5)

thanks to the properties of Hermite and Legendre polynomials, it is easy to check that [14, 16]

$$\begin{aligned} \mathbb {E}\left[ \mathcal {M}_{\ell }\right] =0,\qquad \mathrm{Var}\left( \mathcal {M}_{\ell }\right) =\frac{\log \ell }{32}+O_{\ell \rightarrow +\infty }(1)\text { . } \end{aligned}$$

Moreover, denote by \(\widetilde{\mathcal {L}}_{\ell }(0)\) the standardized nodal length, i.e.

$$\begin{aligned} \widetilde{\mathcal {L}}_{\ell }(0):=\frac{\mathcal {L}_{\ell }(0)-\mathbb {E} [\mathcal {L}_{\ell }(0)]}{\sqrt{\mathrm{Var}(\mathcal {L}_{\ell }(0))}}\text { ,} \end{aligned}$$

and likewise for \(\mathcal {M}_{\ell }\), writing \(\widetilde{\mathcal {M}} _{\ell }:=\frac{\mathcal {M}_{\ell }}{\sqrt{\mathrm{Var}(\mathcal {M}_{\ell })}}.\) It is shown in [19] that, as \(\ell \rightarrow \infty ,\)

$$\begin{aligned} \mathbb {E}\left[ \left| \widetilde{\mathcal {L}}_{\ell }(0)-\widetilde{\mathcal {M}} _{\ell } \right| ^2 \right] =O\left( \frac{1}{\log \ell } \right) \quad \Longrightarrow \quad \widetilde{ \mathcal {L}}_{\ell }(0) =\widetilde{\mathcal {M}}_{\ell }+o_{\mathbb P}(1), \end{aligned}$$
(1.6)

where by \(o_{\mathbb P}(1)\) we mean a sequence of random variables converging to zero in probability. Note that bearing in mind [16, Theorem 1.1], (1.6) yields immediately a CLT for the normalized nodal length.

1.1.2 Boundary Length

A natural further question to investigate is then 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 \(u\in \mathbb R\). Recall that the latter are the area [15, 16], (half) the boundary length, and the Euler–Poincaré characteristic [4] of the excursion regions \(A_u(f_{\ell })=\{x \in \mathbb {S}^2:f_{\ell }(x) \ge u\}\). See Remark 1.2 for an overview.

To this aim, let us first recall that (see [23, 28]), for \(u\in \mathbb R\) and \(\ell \in \mathbb N^*\),

$$\begin{aligned} \mathbb {E}\left[ \mathcal {L}_{\ell }(u)\right] =2\pi \sqrt{\frac{\lambda _{\ell }}{2}}\,\mathrm{e}^{-u^{2}/2}, \end{aligned}$$
(1.7)

and, for \(u\ne 0\), as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \mathrm{Var}\left( \mathcal {L}_{\ell }(u)\right) = \frac{\pi ^2}{2} u^4 \mathrm{e}^{-u^2} \ell \,(1 + o(1)). \end{aligned}$$
(1.8)

Note that for a fixed \(u \ne 0\) and in the regime \(\ell \rightarrow \infty \), the variance of \(\mathcal {L}_{\ell }(u)\) is much larger than in the case \(u=0\). Let us now consider the random variable (recall that \(H_2(t) = t^2 -1\) is the second Hermite polynomial)

$$\begin{aligned} \begin{aligned} \mathcal D_\ell (u) :=&\frac{1 }{4}\sqrt{ \frac{\lambda _{\ell }}{2}}\, u^{2}\mathrm{e}^{-u^2/2} \int _{ \mathbb {S}^{2}}H_{2}(f_{\ell }(x))\,\hbox {d}x. \end{aligned} \end{aligned}$$
(1.9)

Note that \(\mathbb {E}\left[ \mathcal {L}_{\ell }(u)\right] =0\); also, from (1.9) we easily find (cf. (1.8))

$$\begin{aligned} \mathrm{Var}( \mathcal D_\ell (u)) = \pi ^2 \frac{\lambda _\ell }{2\ell +1} u^4 \mathrm{e}^{-u^2}; \end{aligned}$$
(1.10)

let us now consider, for \(u\ne 0\), the standardized level set length, i.e.

$$\begin{aligned} \widetilde{\mathcal {L}}_{\ell }(u):=\frac{\mathcal {L}_{\ell }(u)-\mathbb {E} [\mathcal {L}_{\ell }(u)]}{\sqrt{\mathrm{Var}(\mathcal {L}_{\ell }(u))}}\text { ,} \end{aligned}$$

and analogously \(\widetilde{\mathcal {D}}_{\ell }(u):=\frac{\mathcal {D}_{\ell }(u)}{\sqrt{\mathrm{Var}(\mathcal {D}_{\ell }(u))}}.\) It was shown in [23] that, for \(u\ne 0\), as \(\ell \rightarrow +\infty \),

$$\begin{aligned} {\mathbb { E}}[|\mathcal L_\ell (u) - {\mathbb { E}}[\mathcal L_\ell (u)] - \mathcal D_\ell (u)|^2] =o(\ell ) \quad \Rightarrow \quad \widetilde{\mathcal {L}}_{\ell }(u) =\widetilde{\mathcal {D}}_{\ell }(u)+o_{\mathbb P}(1). \end{aligned}$$
(1.11)

In Sect. 2, we will interpret the results recalled in Sects. 1.1.1 and 1.1.2 for nodal and nonzero level sets lengths, respectively, in terms of so-called chaotic components.

Remark 1.2

(i) 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 \(\ell \rightarrow +\infty \)) perfectly correlated (meaning that the squared correlation coefficients \(\rho (\cdot , \cdot )\) converge to one) with other geometric functionals, such as the area [15, 16] and the Euler–Poincaré characteristic [4] for the excursion regions \(A_u(f_{\ell })=\{x \in \mathbb {S}^2:f_{\ell }(x) \ge u\}\); likewise, perfect correlation holds between these functionals and the number of critical points [5] for \(f_{\ell }(x)\) in the same excursion region, for any non-zero threshold \(u \ne 0\). In other words, at high values of \(\ell \), 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. (ii) From (1.9), it is convenient to note that

$$\begin{aligned} \int _{\mathbb {S}^{2}}H_{2}(f_{\ell }(x))\,\hbox {d}x=\int _{\mathbb {S}^{2}}(f_{\ell }^{2}(x)-1)\,\hbox {d}x=\left\| f_{\ell }\right\| _{L^2(\mathbb S^2)}^{2}-4\pi =\left\| f_{\ell }\right\| _{L^2(\mathbb S^2)}^{2}-\mathbb {E}\left[ \left\| f_{\ell }\right\| _{L^2(\mathbb S^2)}^{2}\right] ; \end{aligned}$$

moreover, a simple application of Parseval’s identity yields

$$\begin{aligned} \left\| f_{\ell }\right\| _{L^2(\mathbb S^2)}^{2}=\frac{4\pi }{2\ell +1}\sum _{m=-\ell }^{\ell } |a_{\ell m}|^{2}=4\pi \,\widehat{C}_{\ell }\text { ,} \end{aligned}$$

where \(\widehat{C}_{\ell }:=(2\ell +1)^{-1}\sum _{m=-\ell }^{\ell } |a_{\ell m}|^{2}\) is usually denoted the sample power spectrum. Roughly speaking, the boundary length is asymptotically proportional to the (random) square norm of the eigenfunctions: from this point of view, it is more natural to expect that it should not play a role in the behaviour of nodal lines, which are clearly invariant to scaling factors.

Note that (ii) in Remarks 1.2 and (1.11) immediately give a CLT for the normalized boundary length.

1.2 Main Result

The main result in this paper is the characterization of the correlation structure between the lengths of the nodal and nonzero level sets. To do so, it is important to recall the standard distinction between the classical correlation coefficient between two (finite-variance) random variables X, Y, which of course is given by

$$\begin{aligned} \rho (X,Y):=\frac{\mathrm{Cov\,}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}\in [-1,1],\quad \mathrm{Cov\,}(X,Y)=\mathbb {E}[XY]-\mathbb {E}[X]\mathbb {E}[Y], \end{aligned}$$

and the Partial Correlation Coefficient (conditional on the finite variance random variable Z), i.e.

$$\begin{aligned} \rho _{Z}(X,Y):=\rho ({X^*},{Y^*}), \end{aligned}$$
(1.12)

where \({X^*},{Y^*}\) are the “residuals” after projecting X, Y on the “explanatory variable” Z, i.e. for a finite variance random variable W

$$\begin{aligned} {W^*}:=(W-\mathbb {E}[W])-\frac{\mathrm{Cov\,}(W,Z)}{\mathrm{Var}(Z)}(Z-\mathbb {E}[Z] \mathbb {)}.\ \end{aligned}$$
(1.13)

As well known, \(\rho _{Z}(X,Y)\) admits a standard interpretation as a measure of the (linear) dependence between X and Y,  after we got rid of the common components depending on Z. Our main result is obtained by taking X, Y to be the boundary lengths at different levels and \(Z=\left\| f_{\ell }\right\| _{L^{2}(\mathbb S^{2})}^{2}=4\pi \,\widehat{C}_{\ell }\) to be the random \(L^{2}\)-norm of the eigenfunctions \(f_{\ell }\):

Theorem 1.3

As \(\ell \rightarrow \infty ,\) for all \(u_{1}\ne 0\)

$$\begin{aligned} \lim _{\ell \rightarrow \infty }\rho \left( \mathcal {L}_{\ell }(u_{1}), \mathcal {L}_{\ell }(u_{2})\right) =\left\{ \begin{array}{c} 1,\qquad \text {if } u_{2}\ne 0, \\ 0,\qquad \text {if } u_{2}=0; \end{array} \right. \end{aligned}$$
(1.14)

on the other hand,

$$\begin{aligned} \lim _{\ell \rightarrow \infty }\rho _{\left\| f_{\ell }\right\| _{L^{2}(\mathbb S^{2})}^{2}}\left( \mathcal {L}_{\ell }(u_{1}),\mathcal {L}_{\ell }(u_{2})\right) =1,\quad \forall u_{1},u_{2}\in \mathbb {R}. \end{aligned}$$
(1.15)

Let us briefly explain the main ideas behind the proof of Theorem 1.3 (for further details see Sect. 2). First, we can interpret (1.6) and (1.11) as a Taylor-type expansion in \(L^2(\mathbb P)\) up to the order 4 and 2, respectively, of \(\widetilde{\mathcal L}_\ell (0)\) and \(\widetilde{\mathcal L}_\ell (u)\) for \(u\ne 0\); Equation 1.14 comes from the latter once recalling orthogonal properties [13, Remark 4.10] of Hermite polynomials (see Sect. 2.1). Removing the effect on \(\mathcal L_\ell (u)\) of the random \(L^2\)-norm of the eigenfunction makes (1.11) useless so we need a further term in the Taylor-type expansion of the boundary length (Propositions 2.3 and 2.4). It turns out that the leading term of the residual \(\widetilde{\mathcal L}_\ell (u)^*\) is “proportional” to the r.v. \(\mathcal M_\ell \) in (1.5) entailing (1.15).

As an easy corollary of Theorem 1.3, we obtain the following joint weak convergence results for the normalized boundary and nodal length. It suffices to recall (1.6) and (1.11), the orthogonality of Wiener chaoses Sect. 2.1 and joint convergence results in [22].

Corollary 1.4

As \(\ell \rightarrow \infty ,\) for all \(u\ne 0\)

$$\begin{aligned} \left( \widetilde{\mathcal {L}}_{\ell }(u),\widetilde{\mathcal {L}}_{\ell }(0)\right) \rightarrow _{d}(Z_{1},Z_{2})\text { ,} \end{aligned}$$

where \((Z_{1},Z_{2})\) denotes a bivariate vector of standard, independent Gaussian variables.

Corollary 1.4 states that the limiting distribution of the nodal and boundary length (at non-zero level) is independent, in the limit for higher and higher eigenvalues. As motivated above, this result is substantially spurious, as it depends crucially on the dominant role played in the boundary length behaviour by the random norm of the eigenfunction. Taking the effect into account, the landscape changes entirely; indeed, consider the “regression residual”

$$\begin{aligned} \mathcal {L}_{\ell |\widehat{C}_{\ell }}(u):=\mathcal {L}_{\ell }(u)-\mathbb {E} [\mathcal {L}_{\ell }(u)]-\frac{\mathrm{Cov\,}(\mathbb {\mathcal {L}_{\ell }}(u), \widehat{C}_{\ell })}{\mathrm{Var}(\widehat{C}_{\ell })}\left( \widehat{C}_{\ell }-\mathbb {E}[\widehat{C}_{\ell }] \right) , \end{aligned}$$
(1.16)

which we again normalize by taking

$$\begin{aligned} \widetilde{\mathcal {L}}_{\ell |\widehat{C}_{\ell }}(u):=\frac{\mathcal {L} _{\ell |\widehat{C}_{\ell }}(u)}{\sqrt{\mathrm{Var}\left( \mathcal {L}_{\ell | \widehat{C}_{\ell }}(u)\right) }}. \end{aligned}$$

We have then the next, fully degenerate, convergence result.

Corollary 1.5

As \(\ell \rightarrow \infty ,\) for all \(u\in \mathbb {R}\)

$$\begin{aligned} \left( \widetilde{\mathcal {L}}_{\ell |\widehat{C}_{\ell }}(u),\widetilde{ \mathcal {L}}_{\ell |\widehat{C}_{\ell }}(0)\right) \rightarrow _{d}(Z,Z) \text { ,} \end{aligned}$$

where Z denotes a standard Gaussian variable.

In short, boundary (nonzero level) and nodal length are asymptotically independent, meaning also that the nodal length carries no information on other functionals such as the excursion area, the Euler–Poincaré characteristic or the number of critical points above a given threshold (see Remark 1.2). This result, however, must be interpreted with great care: it is due to the dominant role played by the sample norm in the behaviour of excursion sets. When this effect is properly subtracted, the behaviour of length fluctuations at any level is fully explained by the nodal length, in the high-energy limit, and the joint distributions are completely degenerate. Thus, indeed the nodal lengths are asymptotically sufficient (in the high-energy limit) to characterize the measure of the boundary at any threshold level, provided that the effect of random fluctuations in the norm is properly taken into account. We refer to [10] for some numerical evidence on these and related issues.

We also note that the driving rationale behind these results is the “long-range” behaviour of the covariance function (1.2), as derived from Hilb’s asymptotics. We hence conjecture that similar results will hold for other models with similar covariance structure; for instance, we expect (partial) correlation to be asymptotically full between level curves of the Berry’s’ random wave model considered e.g. in [21, 27]. Likewise, we expect very similar or analogous results to hold when considering random linear finite combinations of harmonics in neighbouring eigenspaces; this would correspond to a “high frequency” perturbation of the covariance function, with no impact on the asymptotic results.

2 Outline of the Paper and Proof of the Main Result

The main ingredient behind our proofs is a neat series representation (chaotic decomposition) of the length and a consequent careful investigation of its chaotic components. In particular, as briefly anticipated, the results recalled in Sects. 1.1.1 and 1.1.2 can be interpreted in terms of Wiener-Itô theory. Let us first recall the notion of Wiener chaos, restricting ourselves to our specific setting on the sphere (see [20, §2.2] and the references therein for a complete discussion).

2.1 Wiener Chaos

Let us consider the sequence \(\{H_k\}_{k\in \mathbb N}\) of Hermite polynomials on \(\mathbb {R}\); these are defined as follows: \(H_0 \equiv 1\) and

$$\begin{aligned}H_{k}(t) := (-1)^k \phi ^{-1}(t) \frac{\hbox {d}^k}{\hbox {d}t^k}\phi (t), \qquad k\in \mathbb N^*,\end{aligned}$$

where \(\phi (t):=(\sqrt{2\pi })^{-1}\exp (-t^2/2)\) denotes the standard Gaussian density on \(\mathbb R\). The family

$$\begin{aligned} \mathbb {H} := \{(k!)^{-1/2} H_k, k\in \mathbb N\}\end{aligned}$$

is a complete orthonormal system in the space of functions \(L^2(\mathbb R, \mathcal B(\mathbb R), \phi (t)dt)=:L^2(\phi )\). Recall from Sect. 1.1 that the random variables \(\lbrace a_{\ell ,m},m=-\ell ,\dots ,\ell ,\ell \in \mathbb N^*\rbrace \) are defined on the probability space \((\Omega , \mathcal F, \mathbb P)\). Bearing in mind (1.1) and Remark 1.1, we define the space \({\mathcal X}\) to be the closure in \(L^2(\mathbb {P}):=L^2(\Omega , \mathcal F, \mathbb P)\) of all real finite linear combinations of random variables \(\xi \) of the form \(\xi = r \, a_{\ell ,0}\) for some \(r\in \mathbb R\) and \(\ell \in \mathbb N^*\) or for \(m\ne 0\) and some \(\ell \in \mathbb N^*\)

$$\begin{aligned} \xi = z \, a_{\ell ,m} + \overline{z} \, (-1)^m a_{\ell , -m},\qquad z\in \mathbb {C},\end{aligned}$$

thus \({\mathcal X}\) is a real centred Gaussian Hilbert (closed) subspace of \(L^2(\mathbb {P})\). Note that the same space is generated also by the basis elements \(\{a_{\ell ,0},\mathcal {R}(a_{\ell ,m}),\mathcal {I}(a_{\ell ,m}), m=-\ell ,\dots ,\ell , \ell \in \mathbb N^*\}\). Let \(q\ge 0\) be an integer; we define the qth Wiener chaos \(C_q\) associated with \({\mathcal X}\) as the closure in \(L^2(\mathbb {P})\) of all real finite linear combinations of random variables of the type

$$\begin{aligned} H_{p_1}(\xi _1)\cdot H_{p_2}(\xi _2)\cdots H_{p_k}(\xi _k),\qquad k\ge 1, \end{aligned}$$

where \(p_1,...,p_k \in \mathbb N\) are such that \(p_1+\cdots +p_k = q\), and \((\xi _1,...,\xi _k)\) is a standard real Gaussian vector extracted from \({\mathcal X}\); more explicitly the variables \(\xi _i, \text { } i=1,...,k\) are independent Gaussian with unit variance and zero mean. Note that, in particular, \(C_0 = \mathbb {R}\).

The orthonormality and completeness of \(\mathbb {H}\) in \(L^2(\phi )\), together with a standard monotone class argument [20, Theorem 2.2.4] imply that \(C_q \,\bot \, C_m\) in \(L^2(\mathbb {P})\) for every \(q\ne m\), and

$$\begin{aligned} L^2(\Omega , \sigma (\mathcal X), \mathbb {P}) = \bigoplus _{q=0}^\infty C_q, \end{aligned}$$

that is, every square integrable real-valued functional F of \({\mathcal X}\) can be (uniquely) represented as a series, converging in \(L^2(\mathbb P)\), of the form

$$\begin{aligned} F = \sum _{q=0}^\infty \text {proj}[F| q], \end{aligned}$$
(2.1)

where \(\text {proj}[F| q]:=\mathrm{proj}(F \, | \, C_q)\) stands for the projection of F onto \(C_q\) (\(\text {proj}[F| 0]=\mathrm{proj}(F \, | \, C_0) = {\mathbb { E}}[F]\) since \(C_0=\mathbb R\)). From now on, we assume that \(\mathcal F=\sigma (\mathcal X)\), in particular, \(L^2(\Omega , \sigma (\mathcal X), \mathbb {P}) = L^2(\mathbb P)\).

2.2 Chaotic Expansions for Lengths

The perimeter of the boundary of excursion sets on the sphere can be (formally) written as

$$\begin{aligned} \mathcal L_\ell (u) = \int _{\mathbb S^2} \delta _u(f_\ell (x)) \Vert \nabla f_\ell (x)\Vert \,\hbox {d}x \end{aligned}$$
(2.2)

where \(\delta _u\) denotes the Dirac mass in u, \(\nabla f_\ell \) the gradient field and \(\Vert \cdot \Vert \) the Euclidean norm in \(\mathbb {R}^2\). Indeed, let us consider the \(\varepsilon \)-approximating random variable (\(\varepsilon >0\))

$$\begin{aligned} \mathcal L_\ell ^\varepsilon (u) := \frac{1}{2\varepsilon } \int _{\mathbb S^2} 1_{[u-\varepsilon , u+\varepsilon ]}(f_\ell (x))\, \Vert \nabla f_\ell (x)\Vert \, \hbox {d}x, \end{aligned}$$

where \(1_{[u-\varepsilon , u+\varepsilon ]}\) denotes the indicator function of the interval \([u-\varepsilon , u+\varepsilon ]\). We have the following result whose proof is similar to the one given in the nodal case (for details see [19, Appendix B]), and hence omitted.

Lemma 2.1

For \(u\in \mathbb R\),

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \mathcal L_\ell ^\varepsilon (u) = \text { length}(f_\ell ^{-1}(u)), \end{aligned}$$

(cf. (1.3)) both a.s. and in \(L^2({\mathbb {P}})\).

Lemma 2.1 justifies (2.2). By a differentiation of (1.1) of \(f_\ell \), it is easy to see that the random eigenfunctions \(f_\ell \) and the components of \(\nabla f_\ell \), viewed as collections of Gaussian random variables indexed by \(x\in \mathbb S^2\), are all lying in \({\mathcal X}\); hence \(\mathcal L^\varepsilon _\ell (u), \mathcal L_\ell (u)\in L^2(\Omega , \sigma ({\mathcal X}), \mathbb {P})\).

From the chaotic expansion of \(\mathcal L_\ell ^\varepsilon (u)\), it is easy to obtain those of \(\mathcal L_\ell (u)\) by letting \(\varepsilon \) go to zero. In order to recall the chaotic expansion (2.1)

$$\begin{aligned} \mathcal L^\varepsilon _\ell (u) = \sum _{q=0}^{+\infty } \text {proj}[\mathcal L_\ell ^\varepsilon (u) | {q}] \end{aligned}$$
(2.3)

for \(\mathcal L_\ell ^\varepsilon (u)\), let us first write

$$\begin{aligned} \mathcal L_\ell ^\varepsilon (u) =\frac{1}{2\varepsilon } \sqrt{\frac{\ell (\ell +1)}{2}}\int _{\mathbb S^2} 1_{[u-\varepsilon ,u+\varepsilon ]}(f_\ell (x)) \Vert \widetilde{\nabla }f_\ell (x)\Vert \,\hbox {d}x, \end{aligned}$$
(2.4)

where \(\widetilde{\nabla }\) is the normalized gradient, i.e. \(\widetilde{\nabla }:= \nabla / \sqrt{\frac{\ell (\ell +1)}{2}}\). (The variance of each component of \(\nabla f_\ell (x)\) is \(\ell (\ell +1)/2\).) Note that for each \(x\in \mathbb S^2\), the random variables \(f_\ell (x), \nabla f_\ell (x)\) are independent, and the components of \(\nabla f_\ell (x)\) are independent as well.

In [23, Proposition 7.2.2]—see [18, 19] for the nodal case—the terms of the series on the r.h.s. of (2.3) are explicitly given. Let us introduce now two sequences of real numbers \(\lbrace \beta ^\varepsilon _{k}(u)\rbrace _{k=0}^{+\infty }\) and \(\lbrace \alpha _{2n,2m}\rbrace _{n,m=0}^{+\infty }\) corresponding to the chaotic coefficients of the indicator function \(1_{[u-\varepsilon ,u+\varepsilon ]}(\cdot )/(2\varepsilon )\) and the Euclidean norm, respectively: for \(k=0, 1, 2, \dots \),

$$\begin{aligned} \begin{aligned} \beta ^\varepsilon _{k}(u)&:= \frac{1}{2\varepsilon } \int _{u-\varepsilon }^{u+\varepsilon } H_k(t)\phi (t)\,\hbox {d}t \ \mathop {\longrightarrow }^{\varepsilon \rightarrow 0}\ \phi (u) H_k(u) =: \beta _k(u), \end{aligned} \end{aligned}$$
(2.5)

where \(\phi \) still denotes the standard Gaussian density, while for \(n,m=0, 1, 2, \dots \)

$$\begin{aligned} \alpha _{2n,2m}:=\sqrt{\frac{\pi }{2}}\frac{(2n)!(2m)!}{n!m!}\frac{1}{2^{n+m}}p_{n+m}\left( \frac{1}{4}\right) , \end{aligned}$$
(2.6)

\(p_{N}\) denoting the swinging factorial coefficient

$$\begin{aligned} p_{N}(x):=\sum _{j=0}^{N}(-1)^{j}(-1)^{N}\left( \begin{array}{c} N \\ j \end{array} \right) \frac{(2j+1)!}{(j!)^{2}}x^{j}. \end{aligned}$$

Note that, if \(u=0\), the coefficient \(\beta ^\varepsilon _{k}(0)=0\) (and hence \(\beta _k(0)=0\)) whenever k is odd.

Proposition 2.2

The chaotic expansion (2.1) of the approximate length is

$$\begin{aligned} \begin{aligned} \mathcal {L}_{\ell }^\varepsilon (u)&=\sum _{q=0}^{+\infty } \text {proj}[\mathcal L_\ell ^\varepsilon (u)| {q}] = \sum _{q=0}^{\infty }\sqrt{\frac{\ell (\ell +1)}{2}}\sum _{2a + 2b + c = q}\frac{\alpha _{2a,2b}\beta _{c}^\varepsilon (u)}{(2a)!(2b)!c!}\\&\quad \times \int _{\mathbb {S}^{2}}H_{c}(f_{\ell }(x))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(x))H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(x))\,\hbox {d}x, \end{aligned} \end{aligned}$$
(2.7)

where we use spherical coordinates (colatitude \(\theta ,\) longitude \(\varphi \)) and for \(x=(\theta _{x},\varphi _{x})\) we are using the notation

$$\begin{aligned} \widetilde{\partial }_{1;x}=\left. (\ell (\ell +1)/2)^{-1/2}\cdot \frac{\partial }{\partial \theta }\right| _{\theta =\theta _{x}},\quad \widetilde{\partial }_{2;x}= (\ell (\ell +1)/2)^{-1/2}\cdot \left. \frac{1}{\sin \theta } \frac{\partial }{\partial \varphi }\right| _{\theta =\theta _{x},\varphi , =\varphi _{x}}, \end{aligned}$$

and the chaotic coefficients are as in (2.5) and (2.6). By letting \(\varepsilon \rightarrow 0\), we find

$$\begin{aligned} \begin{aligned} \mathcal {L}_{\ell }(u) =\sum _{q=0}^{+\infty } \text {proj}[\mathcal L_\ell (u)| {q}]&= \sum _{q=0}^{\infty }\sqrt{\frac{\ell (\ell +1)}{2}}\sum _{2a + 2b + c = q}\frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!}\\&\quad \times \int _{\mathbb {S}^{2}}H_{c}(f_{\ell }(x))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(x))H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(x))\,\hbox {d}x, \end{aligned} \end{aligned}$$
(2.8)

where the convergence of the above series is in \(L^2(\mathbb P)\).

Equations 2.7 and 2.8 are in some sense the major formulae of the article; all further proofs are obtained by estimating the various terms in this formulae. The proof of Proposition 2.2 is analogous to the one given in the nodal case (see [19, §2] and [23, Proposition 7.2.2]) and hence omitted; in short, Eq. 2.7 follows from the evaluation of the projection coefficients for any level \(u \in \mathbb {R}\), whereas in the previous references the computation was only considered for \(u = 0\); Eq. 2.8 follows from Eq. 2.7 and Lemma 2.1.

Recall now from Sect. 1.1 that \(\lambda _\ell = \ell (\ell +1)\). As it will be clearer later, it suffices to deal with the first few terms of the series in (2.8). For every \(u\in \mathbb R\), from (2.8), (2.6) and (2.5)

$$\begin{aligned} \text {proj}[\mathcal {L}_{\ell }(u)|0]=\sqrt{ \frac{\lambda _{\ell }}{2}} \alpha _{0,0} \beta _{0}(u) \int _{\mathbb S^2} H_0(f_\ell (x))\,\hbox {d}x= \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\phi (u) \cdot 4\pi = \mathbb E[\mathcal L_\ell (u)], \end{aligned}$$
(2.9)

cf. (1.7),

$$\begin{aligned} \text {proj}[\mathcal {L}_{\ell }(u)|1] = 0 \end{aligned}$$
(2.10)

since spherical harmonics have zero mean on the sphere, and thanks to Green formula (see [25, §4] )

$$\begin{aligned} \text {proj}[\mathcal {L}_{\ell }(u)|2]=\sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}} u^{2}\phi (u) \frac{1}{2}\int _{\mathbb {S}^{2}}H_{2}(f_{\ell }(x))\,\hbox {d}x, \end{aligned}$$
(2.11)

that is, \(\mathcal D_\ell (u)\) (cf. (1.9)); notice that the right-hand side of (2.11) is identically equal to zero if and only if \(u=0\) (see also [19]). Obviously, if \(u=0\), the term \(\text {proj}[\mathcal {L}_{\ell }(0)|q]\) vanishes whenever q is odd.

We are in a position to make some more comments, comparing results recalled in Sects. 1.1.1 and 1.1.2, and the theory of chaotic decompositions exploited above: the leading term in the \(L^{2}(\mathbb P)\)-expansion of the boundary length around its expected value is provided by its orthogonal projection on the second-order Wiener chaos (2.11), rather than the fourth as in the nodal case (see (1.5) and (1.6), and compare with Proposition 2.4); the asymptotic variance of the second chaos is of order \(\ell \), as opposed to \(\log \ell \) for the fourth-order chaos (see e.g. [16, 19]), and hence, the variance of nonzero level curves is larger than in the nodal case (the Berry’s cancellation phenomenon, see [28]). Similar to the nodal length case (1.5), the projection (2.11) takes a very simple form, as the integral depends only on the (second power of the) random eigenfunction and not on its gradient.

2.3 Proof of the Main Result

In this paper, we complete the characterization of the chaos expansion for the boundary length of excursion sets, and indeed, we show the following results which are of independent interest.

Proposition 2.3

For all \(u\in \mathbb {R},\) as \(\ell \rightarrow \infty \)

$$\begin{aligned} \mathrm{Var}\left( \sum _{q=3}^{+\infty } \text { proj}[\mathcal L_\ell (u)| q] \right) = \frac{\pi }{4} \phi (u)^2 \left( H_{4}(u)+2H_{2}(u)-\frac{3}{2}\right) ^2 \log \ell + O(1). \end{aligned}$$

Of course, the component in the series is orthogonal by construction, so the variance is just the sum of their variances; for \(u=0\), from Propositions 2.2 and 2.3, it gives an alternative proof for (1.4). Moreover we get, for \(u\ne 0\),

$$\begin{aligned} \mathrm{Var}(\mathcal L_\ell (u)) = \pi ^2 \frac{\lambda _\ell }{2\ell +1} u^4 \mathrm{e}^{-u^2} + O_{\ell \rightarrow +\infty } (\log \ell ) \end{aligned}$$
(2.12)

(cf. (1.8)). Let us set for notational simplicity

$$\begin{aligned} \mathcal M_\ell (u) := \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!}\int _{\mathbb {S}^{2}}H_{4}(f_{\ell }(x))\hbox {d}x. \end{aligned}$$

where

$$\begin{aligned} \mathcal {G}(u):=\phi (u)\left( H_{4}(u)+2H_{2}(u)-\frac{3}{2}\right) , \end{aligned}$$

so that \(\mathcal M_\ell (0) = \mathcal M_\ell \) in (1.5). As remarked earlier, the asymptotic variance of \(\int _{\mathbb {S}^{2}}H_{4}(f_{\ell }(x))\hbox {d}x\) is known (see [16], Lemma 3.2); indeed, we have

$$\begin{aligned} \mathbb {E}\left[ \int _{\mathbb {S}^{2}}H_{4}(f_{\ell }(x))\hbox {d}x \right] ^2=576\frac{\log \ell }{\ell ^2}+O\left( \frac{1}{\ell ^2}\right) , \end{aligned}$$

and it is hence easy to show that, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \mathrm{Var}\left( \mathcal M_\ell (u) \right) = \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell + O\left( 1 \right) . \end{aligned}$$
(2.13)

Proposition 2.4

For all \(u\in \mathbb {R},\) as \(\ell \rightarrow \infty \),

$$\begin{aligned} \rho \left( \sum _{q=3}^{+\infty } \text { proj}[\mathcal L_\ell (u)| q], \mathcal M_\ell (u) \right) = 1 + O\left( \frac{1}{\log \ell } \right) . \end{aligned}$$

Note that Proposition 2.4 generalizes results in [19], moreover implies that

$$\begin{aligned} \mathrm{Cov\,}(\text {proj}[\mathcal L_\ell (u)|4], \mathcal M_\ell (u)) \sim \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell . \end{aligned}$$
(2.14)

Here we are also exploiting the fact that the terms \(\text {proj}[\mathcal L_\ell (u)| q]\) for \(q \ne 4\) are orthogonal to \(\mathcal M_\ell (u)\), which is an element of the fourth-order chaos. Applying Cauchy–Schwartz inequality to the l.h.s. of (2.14), thanks to orthogonality of Wiener chaoses Sect. 2.1, Propositions 2.3, (2.13) and (2.14), we have that, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \mathrm{Var}\left( \text {proj}[\mathcal L_\ell (u)|4] \right) \sim \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell , \end{aligned}$$
(2.15)

so the contribution of the variance of \(\text {proj}[\mathcal L_\ell (u)|3] + \sum _{q=5}^{+\infty } \text {proj}[\mathcal L_\ell (u)| q]\) is negligible with respect to \(\mathrm{Var}\left( \text {proj}[\mathcal L_\ell (u)|4] \right) \), and moreover, we get the following.

Corollary 2.5

For all \(u\in \mathbb {R},\) as \(\ell \rightarrow \infty \)

$$\begin{aligned} \rho \left( \text { proj}[\mathcal L_\ell (u)| 4], \mathcal M_\ell (u) \right) = 1 + O\left( \frac{1}{\log \ell } \right) . \end{aligned}$$

The proofs of Propositions 2.3 and 2.4 are technical and will be given in the next sections. We are now ready to prove our main result.

Proof of Theorem 1.3

Let us consider \(u_1, u_2\in \mathbb R\); recall that

$$\begin{aligned} \rho ({\mathcal L}_\ell (u_1), {\mathcal L}_\ell (u_2)) = \frac{\mathrm{Cov\,}(\mathcal L_\ell (u_1), \mathcal L_\ell (u_2))}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1))\mathrm{Var}(\mathcal L_\ell (u_2))}}. \end{aligned}$$

Assume first that \(u_1 \cdot u_2 \ne 0\), thanks to Propositions 2.2 and 2.3

$$\begin{aligned} \mathcal L_\ell (u_i) - {\mathbb { E}}[\mathcal L_\ell (u_i) ]= \text {proj}[\mathcal L_\ell (u_i)|2] + \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u_i)|q]=\mathcal D_\ell (u_i) + \mathcal R_\ell (u_i), \end{aligned}$$
(2.16)

where \(\mathrm{Var}(\mathcal R_\ell (u_i)) = O(\log \ell )\), as \(\ell \rightarrow +\infty \). We have

$$\begin{aligned} \rho ({\mathcal L}_\ell (u_1), {\mathcal L}_\ell (u_2))= & {} \frac{\mathrm{Cov\,}(\mathcal D_\ell (u_1) + \mathcal R_\ell (u_1), \mathcal D_\ell (u_2) + \mathcal R_\ell (u_2))}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}\nonumber \\= & {} \frac{\mathrm{Cov\,}(\mathcal D_\ell (u_1), \mathcal D_\ell (u_2) )}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}\nonumber \\&+ \frac{\mathrm{Cov\,}(\mathcal D_\ell (u_1), \mathcal R_\ell (u_2) ) + \mathrm{Cov\,}(\mathcal D_\ell (u_2), \mathcal R_\ell (u_1) )}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}\nonumber \\&+ \frac{\mathrm{Cov\,}(\mathcal R_\ell (u_1), \mathcal R_\ell (u_2) )}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}. \end{aligned}$$
(2.17)

Let us bear in mind also (1.10), (2.12) and (2.16). Thanks to Cauchy–Schwartz inequality applied to the last two terms on the r.h.s. of (2.17), we find, as \(\ell \rightarrow +\infty \),

$$\begin{aligned}&\frac{ \left| \mathrm{Cov\,}(\mathcal D_\ell (u_1), \mathcal R_\ell (u_2) ) + \mathrm{Cov\,}(\mathcal D_\ell (u_2), \mathcal R_\ell (u_1) ) \right| }{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}} = O\left( \sqrt{\frac{\log \ell }{\ell }}\right) , \nonumber \\&\frac{\left| \mathrm{Cov\,}(\mathcal R_\ell (u_1), \mathcal R_\ell (u_2) ) \right| }{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))} }= O\left( \frac{\log \ell }{\ell }\right) .\nonumber \\ \end{aligned}$$
(2.18)

For the first term on the r.h.s. of (2.17), we have

$$\begin{aligned} \frac{\mathrm{Cov\,}(\mathcal D_\ell (u_1), \mathcal D_\ell (u_2) )}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}= & {} \frac{1}{(4\pi )^2}\frac{2\ell +1}{2}\mathrm{Var}\left( \int _{\mathbb S^2} H_2(f_\ell (x))\,\hbox {d}x\right) \nonumber \\&\times \frac{\pi ^2 \frac{\lambda _\ell }{2\ell +1}u_1^2 u_2^2 \text {e}^{-u_1^2/2}\text {e}^{-u_2^2/2}}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}\nonumber \\= & {} \frac{1}{(4\pi )^2}\frac{2\ell +1}{2} 8\pi ^2 \cdot 2 \int _{-1}^1 P_\ell (t)^2\,dt\nonumber \\&\times \frac{\pi ^2 \frac{\lambda _\ell }{2\ell +1}u_1^2 u_2^2 \text {e}^{-u_1^2/2}\text {e}^{-u_2^2/2}}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}}\nonumber \\= & {} \frac{\pi ^2 \frac{\lambda _\ell }{2\ell +1}u_1^2 u_2^2 \text {e}^{-u_1^2/2}\text {e}^{-u_2^2/2}}{\sqrt{\mathrm{Var}(\mathcal L_\ell (u_1)) \mathrm{Var}(\mathcal L_\ell (u_2))}} \nonumber \\= & {} 1 + O_{\ell \rightarrow +\infty }\left( \frac{\log \ell }{\ell }\right) , \end{aligned}$$
(2.19)

where we used (1.9), (2.12) and standard properties of Legendre and Hermite polynomials. Now plugging (2.18) and (2.19) into (2.17), we get what we were looking for i.e. as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \rho (\mathcal L_\ell (u_1), \mathcal L_\ell (u_2)) = 1 + O\left( \sqrt{\frac{\log \ell }{\ell }}\right) . \end{aligned}$$
(2.20)

For the nodal length we have, from (1.6) (see also Proposition 2.2 and Corollary 2.5)

$$\begin{aligned} \mathcal L_\ell (0) - {\mathbb { E}}[\mathcal L_\ell (0) ] = \text {proj}[\mathcal L_\ell (0)|4] + \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (0)|2q] = \mathcal M_\ell + \mathcal E_\ell , \end{aligned}$$
(2.21)

where \(\mathcal M_\ell \) is as in (1.5) and \(\mathrm{Var}(\mathcal E_\ell ) = O(1)\) as \(\ell \rightarrow +\infty \). Repeating the same argument as for the proof of (2.20), from (2.21) and (2.16) we have, for \(u_1\ne 0\) and as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \rho ({\mathcal L}_\ell (u_1), {\mathcal L}_\ell (0)) = O\left( \frac{1}{\sqrt{\log \ell }}\right) , \end{aligned}$$

which together with (2.20) gives (1.14).

Recall now (1.12) and (1.13), here \(Z=\Vert f_\ell \Vert ^2_{L^2(\mathbb S^2)}\). From Propositions 2.2, (2.11), (1.9) and Remark 1.2, we have, for every \(u\in \mathbb R\) and as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \begin{aligned} \mathcal L_\ell (u)^* =&\sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u)|q]. \end{aligned} \end{aligned}$$
(2.22)

Analogously to the proof of (2.20), from Propositions 2.3 and 2.4 we have, for \(u_1,u_2 \in \mathbb R\),

$$\begin{aligned} \rho _{\Vert f_\ell \Vert ^2_{L^2(\mathbb S^2)}}(\mathcal L_\ell (u_1), \mathcal L_\ell (u_2)) = 1 + O\left( \frac{1}{\sqrt{\log \ell }}\right) \end{aligned}$$

which proves (1.15). The proof of Theorem 1.3 is hence complete. \(\square \)

2.4 Discussion

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-universal) non-central limit theorem for arithmetic random waves, i.e. Gaussian Laplacian eigenfunctions on the standard two-dimensional flat torus \(\mathbb {T}^{2}:=\mathbb R^2/\mathbb Z^2,\) was established in [18]. To obtain this result, analogously to our discussion above the nodal length was decomposed into chaotic components (see Sect. 2.2). The expansion of nodal length in the toroidal and spherical cases has both analogies and important differences. In both cases, the term corresponding to \(q=2\) disappears at \(u=0,\) thus entailing that the variance becomes of lower order (the so-called Berry’s cancellation phenomenon). Likewise, in both cases the nodal length is dominated by the fourth-order chaos: however, it is only in the spherical case that the fourth-order term admits an expression depending on the field only (and not on the gradient components). Because of this, we do not expect that taking into account the random norm behaviour will be enough to establish full correlation between nodal length and boundary curves (it could be the case that a degeneracy occurs when a sufficient number of different levels is considered).

Similar cancellation phenomena occur for other geometric functionals, including the excursion area and the Defect ([15,16,17, 24], which cover any dimension \(d\ge 2\)), the Euler–Poincaré characteristic [4] and the zeros of complex arithmetic random waves [8]; quantitative central limit theorems have been given on the sphere in [4, 15,16,17] for many of these statistics, in the high-energy limit where \(\ell \rightarrow \infty .\) On the torus, the asymptotic behaviour has been shown to be more complicated, because it is non-Gaussian and differs across different subsequences as the eigenvalue diverges (see [8, 18]).

3 Proof of Proposition 2.3

Let us bear in mind Proposition 2.2. For \(q\ge 0, q\ne 2\) we set

$$\begin{aligned} \Psi ^\varepsilon _\ell (x;q):= & {} \sqrt{\frac{\lambda _\ell }{2}}\sum _{2a + 2b + c = q}\frac{\alpha _{2a,2b}\beta _{c}^\varepsilon (u)}{(2a)!(2b)!c!} H_{c}(f_{\ell }(x))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(x))H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(x)),\nonumber \\ \Psi _\ell (x;q):= & {} \sqrt{\frac{\lambda _\ell }{2}}\sum _{2a + 2b + c = q}\frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!} H_{c}(f_{\ell }(x))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(x))H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(x)).\nonumber \\ \end{aligned}$$
(3.1)

These two functions can be viewed as the qth order component in the integrand of the chaos expansion given in the proposition; they are both polynomial functions in \(f_{\ell }\) and its first-order derivatives, and they depend on the parameter u (although we do not take this into account to simplify our notation, this does not affect the discussion to follow).

Concerning second chaotic components, thanks to Green’s formula (for details see again [23, 25]), we can write

$$\begin{aligned} \text {proj}[\mathcal L_\ell ^\varepsilon (u)|2] =\sqrt{\frac{\ell (\ell +1)}{2} } \left( \frac{1}{2} \beta ^\varepsilon _{2}(u) \alpha _{0,0} + \beta ^\varepsilon _0(u) \alpha _{2,0} \right) \int _{\mathbb S^2} H_2(f_\ell (x))\,\hbox {d}x\nonumber \\ \end{aligned}$$
(3.2)

(note that \(\text {proj}[\mathcal L_\ell (u) |2] = \mathcal D_\ell (u)\) in (1.9)). Let us hence set

$$\begin{aligned} \Psi _\ell ^\varepsilon (x; 2):= & {} \sqrt{\frac{\ell (\ell +1)}{2} } \left( \frac{1}{2} \beta ^\varepsilon _{2}(u) \alpha _{0,0} + \beta ^\varepsilon _0(u) \alpha _{2,0} \right) H_2(f_\ell (x))\nonumber \\ \Psi _\ell (x; 2):= & {} \sqrt{\frac{\ell (\ell +1)}{2} } \left( \frac{1}{2} \beta _{2}(u) \alpha _{0,0} + \beta _0(u) \alpha _{2,0} \right) H_2(f_\ell (x)) \end{aligned}$$
(3.3)

and finally

$$\begin{aligned} \Psi _\ell ^\varepsilon (x):= & {} \sum _{q=0}^{+\infty } \Psi ^\varepsilon _\ell (x;q),\qquad \Psi _\ell (x) :=\sum _{q=0}^{+\infty } \Psi _\ell (x;q), \end{aligned}$$
(3.4)

where the second equality is just formal; by this we mean that the first series converges in \(L^2(\mathbb {P})\) for every fixed x, while the second does not. Before we proceed, we need to introduce some more notation: let us fix \(\overline{x} = (0,0)\) to be the “north pole” and \(y(\theta ) =(0,\theta )\) to be points on the meridian where \(\varphi =0\). We will split the proof of Proposition 2.3 into some lemmas.

Lemma 3.1

For \(C>0\) large enough

$$\begin{aligned} \begin{aligned} \mathrm{Var}\left( \sum _{q=3}^{+\infty } \text { proj}[\mathcal L_\ell (u) | q] \right) = O(1) + 2\cdot 8\pi ^2 \sum _{q=3}^{+\infty } \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \, \hbox {d}\theta , \end{aligned} \end{aligned}$$

where the constant involved in the O-notation does not depend on \(\ell \).

Let us deal with the terms of the series on the r.h.s. of (3.1).

Lemma 3.2

For \(C>0\), we have

$$\begin{aligned} \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};3) \Psi _\ell (y(\theta );3) \right] \sin \theta \,\hbox {d}\theta = O(1),\qquad \text {as } \ell \rightarrow +\infty . \end{aligned}$$
(3.5)

Lemma 3.3

For \(C>0\), we have

$$\begin{aligned} 2\cdot 8\pi ^2 \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};4) \Psi _\ell (y(\theta );4) \right] \sin \theta \hbox {d}\theta= & {} \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell \nonumber \\&+ O_{\ell \rightarrow +\infty }(1). \end{aligned}$$
(3.6)

Lemma 3.4

For \(C>0\) large enough, we have

$$\begin{aligned} \sum _{q=5}^{+\infty } \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \,\hbox {d}\theta =O_{\ell \rightarrow +\infty }(1). \end{aligned}$$
(3.7)

The proofs of Lemmas 3.2–3.4 are technical and will be given below; the proof of Lemma 3.1 is postponed to Appendix 3.1. Let us now prove Proposition 2.3.

Proof of Proposition 2.3

From (3.1), we have, thanks to Lemmas 3.2–3.4,

$$\begin{aligned} \mathrm{Var}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u) | q] \right)= & {} 2\cdot 8\pi ^2 \sum _{q=3}^{+\infty } \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \, \hbox {d}\theta + O(1)\\= & {} \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell + O_{\ell \rightarrow +\infty }(1) \end{aligned}$$

that immediately concludes the proof. \(\square \)

3.1 Moments of Legendre Polynomials

In order to prove Lemmas 3.2–3.4, we will need some estimates on moments of Legendre polynomials and their derivatives. Let us bear in mind [19, (A1)-(A6)].

Lemma 3.5

For any a, b, c, d nonnegative integers s.t. \(a+b+c+d\ge 3\) and \(C>0\), as \(\ell \rightarrow +\infty \), (\(L:=\ell +\frac{1}{2}\))

$$\begin{aligned}&\left( \frac{2}{\ell (\ell +1)} \right) ^{b + \frac{c}{2} + d} \int _{C/\ell }^{\pi /2} P_\ell (\cos \theta )^a P_\ell '(\cos \theta )^b (P'_\ell (\cos \theta ) \sin \theta )^c \nonumber \\&\qquad \times (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^d \sin \theta \,\hbox {d}\theta \nonumber \\&= \left( \sqrt{\frac{2}{\pi }}\right) ^{a+b+c+d} \frac{2^{b + \frac{c}{2} + d}}{L^{1+\frac{a+3b+c+d}{2}}} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\sin \psi /L}} \right) ^{a + 3b + c +d -2}\nonumber \\&\quad \sin ^{a+d}(\psi + \pi /4) \sin ^{b +c} (\psi - \pi /4)\,\hbox {d}\psi + O\left( \frac{1}{L^2} \right) . \end{aligned}$$
(3.8)

Proof

The proof relies on a precise asymptotic analysis of Legendre polynomials and their derivatives. From [28, Lemma B.3] (see also [6, §5.3]), we have, uniformly for \(C<\psi < L \pi /2\) \(\left( L:= \ell +1/2 \right) \),

$$\begin{aligned} P_\ell (\cos \psi /L)= & {} \sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{\ell \sin \psi /L}} \left( \sin (\psi + \pi /4) + O(1/\psi ) \right) \nonumber \\ P'_\ell (\cos \psi /L)= & {} \sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{\ell \sin ^3 \psi /L}} \left( \ell \sin (\psi - \pi /4) + O(1) \right) \nonumber \\ P_\ell ^{''} (\cos \psi /L)= & {} - \frac{\ell ^2}{\sin ^2\psi /L} P_\ell (\cos \psi /L) + \frac{2}{\sin ^2 \psi /L}P_\ell '(\cos \psi /L) + O(\ell ^3/\psi ^{5/2}).\nonumber \\ \end{aligned}$$
(3.9)

Some manipulations of (3.9) give (also recalling that \(\psi = O(L)\))

$$\begin{aligned}&P_\ell (\cos \psi /L) = \sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{\ell \sin \psi /L}} \left( \sin (\psi + \pi /4) + O(1/\psi ) \right) \nonumber \\&\quad \frac{2}{\ell (\ell +1)}P'_\ell (\cos \psi /L) = 2\sqrt{\frac{2}{\pi }} \frac{1}{\ell \sqrt{\ell \sin ^3 \psi /L}} \left( \sin (\psi - \pi /4) + O(1/\psi ) \right) \nonumber \\&\quad \sqrt{ \frac{2}{\ell (\ell +1)}} P_\ell '(\cos \psi /L) \sin \psi /L = \sqrt{2}\sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{\ell \sin \psi /L}} \left( \sin (\psi - \pi /4) + O(1/\psi ) \right) \nonumber \\&\quad \frac{2}{\ell (\ell +1)}(P'_\ell (\cos \psi /L)\cos \psi /L - P_\ell ^{''}(\cos \psi /L)\sin ^2\psi /L) = 2 \sqrt{\frac{2}{\pi }} \frac{1}{\sqrt{\ell \sin \psi /L}}\nonumber \\&\quad \Big ( \sin (\psi + \pi /4) \ + O (1/\psi )\Big ).\nonumber \\ \end{aligned}$$
(3.10)

Now plugging (3.10) into the l.h.s. of (3.8), using that for positive integer j

$$\begin{aligned} \left( \sin (\psi \pm \pi /4) + O(1/\psi )\right) ^j = \sin ^j(\psi \pm \pi /4) + O(1/\psi ), \end{aligned}$$

we get

$$\begin{aligned}&\left( \frac{2}{\ell (\ell +1)} \right) ^{b + \frac{c}{2} + d} \int _{C/\ell }^{\pi /2} P_\ell (\cos \theta )^a P_\ell '(\cos \theta )^b\\&\qquad (P'_\ell (\cos \theta ) \sin \theta )^c (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^d \sin \theta \,\hbox {d}\theta \\&\quad = \left( \sqrt{\frac{2}{\pi }}\right) ^{a+b+c+d} 2^{b + \frac{c}{2} + d}\frac{1}{L} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\ell \sin \psi /L}} \right) ^{a + 3b + c +d} \\&\qquad \times \left( \sin ^{a+d}(\psi + \pi /4) \sin ^{b +c} (\psi - \pi /4) + O(1/\psi ) \right) \sin \psi /L\,\hbox {d}\psi \\&\quad = \left( \sqrt{\frac{2}{\pi }}\right) ^{a+b+c+d} \frac{2^{b + \frac{c}{2} + d}}{L^{1+\frac{a+3b+c+d}{2}}} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\sin \psi /L}} \right) ^{a + 3b + c +d -2}\\&\quad \sin ^{a+d}(\psi + \pi /4) \sin ^{b +c} (\psi - \pi /4)\,\hbox {d}\psi \\&\qquad + O\left( \frac{1}{L^2} \int _C^{L\pi /2} \left( \frac{1}{\psi } \right) ^{\frac{a+3b+c+d}{2}}\,\hbox {d}\psi \right) \end{aligned}$$

which gives (3.8), the integral in the latter error term being convergent (recall in particular that \(a+b+c+d\ge 3\)). \(\square \)

We are now ready to state and prove the results we need for Lemmas  3.2 and 3.3.

Lemma 3.6

For any \(C>0\),

$$\begin{aligned}&\frac{\ell (\ell +1)}{2} \int _{C/\ell }^{\pi /2} P_\ell (\cos \theta )^3 \sin \theta \, \hbox {d}\theta = O_{\ell \rightarrow +\infty }(1);\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta ) (P'_\ell (\cos \theta ) \sin \theta )^2 \sin \theta \,\hbox {d}\theta = O_{\ell \rightarrow +\infty }(1);\nonumber \\&\quad \frac{2}{\ell (\ell +1)} \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta ) (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2\sin \theta \,\hbox {d}\theta = O_{\ell \rightarrow +\infty }(1);\nonumber \\&\quad \frac{2}{\ell (\ell +1)} \int _{C/\ell } ^{\pi /2}(P'_\ell (\cos \theta ) \sin \theta )^2 (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta ) \,\sin \theta \,\hbox {d}\theta = O_{\ell \rightarrow +\infty }(1);\nonumber \\&\quad \frac{2}{\ell (\ell +1)} \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta )(P'_\ell (\cos \theta ))^2 \sin \theta \,\hbox {d}\theta = O_{\ell \rightarrow +\infty }(1). \end{aligned}$$
(3.11)

Proof

Lemma 3.5 entails that the asymptotic behaviour of the first four integrals (note the multiplicative factor \(\ell (\ell +1)/2\)) on the l.h.s. of (3.11) is given by (up to constants)

$$\begin{aligned} \frac{1}{\sqrt{L}} \int _C^{L\pi /2} \frac{1}{\sqrt{\sin \psi /L}} \sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)\,\hbox {d}\psi + O\left( 1 \right) , \end{aligned}$$

where \(a,c,d\in \lbrace 0,1,2\rbrace \) and \(a+c+d=3\). It remains to prove that, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \frac{1}{\sqrt{L}} \int _C^{L\pi /2} \frac{1}{\sqrt{\sin \psi /L}} \sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)\,\hbox {d}\psi = O(1). \end{aligned}$$
(3.12)

This follows by integration by parts with the bounded function

$$\begin{aligned} f(\psi ):=\int _0^\psi \sin ^{a+d} (x +\pi /4) \sin ^{c} (x -\pi /4)\,\hbox {d}x; \end{aligned}$$

indeed, for the l.h.s. of (3.12), as \(\ell \rightarrow +\infty \),

$$\begin{aligned}&\frac{1}{\sqrt{L}} \int _C^{L\pi /2} \frac{1}{\sqrt{\sin \psi /L}} \sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)\,\hbox {d}\psi \\&\quad = \frac{1}{\sqrt{L}} \left( \frac{f(\psi )}{\sqrt{\sin \psi /L}}{\Big |_C^{L\pi /2}} +\frac{1}{2} \int _C^{L\pi /2} \frac{f(\psi )}{\sqrt{\sin ^3 \psi /L}} \frac{\cos \psi /L}{L} \,\hbox {d}\psi \right) \\&\quad = O(1) + O\left( \int _C^{L\pi /2} \frac{1}{\psi ^{3/2}} \right) = O(1) \end{aligned}$$

establishing (3.12). The last integral of (3.11) is simpler, actually by Lemma 3.5

$$\begin{aligned}&\frac{2}{\ell (\ell +1)} \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta )(P'_\ell (\cos \theta ))^2 \sin \theta \,\hbox {d}\theta \\&\quad = \left( \frac{2}{\pi }\right) ^{3/2} \frac{2}{L^{5/2}} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\sin \psi /L}} \right) ^{5} \sin (\psi + \pi /4) \sin ^{2} (\psi - \pi /4)\,\hbox {d}\psi + O\left( 1 \right) \\&\quad = O\left( \int _C^{L\pi /2} \frac{1}{\psi ^{5/2}} \,\hbox {d}\psi \right) + O\left( 1 \right) = O(1). \end{aligned}$$

The proof of Lemma 3.6 is hence complete. \(\square \)

Lemma 3.7

For any \(C>0\), as \(\ell \rightarrow +\infty \)

$$\begin{aligned}&\int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta )^4 \sin \theta \,\hbox {d}\theta = \frac{4}{\pi ^2} \frac{3}{8} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ; \nonumber \\&\quad \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta )^2 \frac{2}{\ell (\ell +1)} (P'_\ell (\cos \theta )\nonumber \\&\quad \sin \theta )^2 \sin \theta \,\hbox {d}\theta = \frac{1}{\pi ^2} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} P_\ell ^2(\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2 \sin \theta \nonumber \\&\hbox {d}\theta = \frac{16}{\pi ^2} \frac{3}{8} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} P_\ell (\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} (P'_\ell (\cos \theta ) \sin \theta )^2 (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )\sin \theta \,\hbox {d}\theta \nonumber \\&\quad = \frac{16}{\pi ^2} \frac{1}{8} \frac{\log \ell }{\ell ^2}+ O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{4}{\ell ^2(\ell +1)^2}P'_\ell (\cos \theta )^4 \sin ^5 \theta \,\hbox {d}\theta = \frac{16}{\pi ^2} \frac{3}{8} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{8}{\ell ^3(\ell +1)^3}(P'_\ell (\cos \theta )\sin \theta )^2 (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2 \nonumber \\&\quad \sin \theta \,\hbox {d}\theta = \frac{32}{\pi ^2} \frac{1}{8} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{2^4}{\ell ^4(\ell +1)^4} (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^4 \sin \theta \,\hbox {d}\theta = \frac{4\cdot 2^4}{\pi ^2} \frac{3}{8} \frac{\log \ell }{\ell ^2} + O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} P_\ell ^2(\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} P'_\ell (\cos \theta )^2 \sin \theta \,\hbox {d}\theta = O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{2}{\ell (\ell +1)} (P'_\ell (\cos \theta ) \sin \theta )^2 \frac{4}{\ell ^2 (\ell +1)^2} P'_\ell (\cos \theta )^2\sin \theta \,\hbox {d}\theta = O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{4}{\ell ^2(\ell +1)^2}(P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2\nonumber \\&\frac{4}{\ell ^2 (\ell +1)^2} P'_\ell (\cos \theta )^2 \sin \theta \,\hbox {d}\theta = O\left( \frac{1}{\ell ^2}\right) ;\nonumber \\&\quad \int _{C/\ell } ^{\pi /2} \frac{2^4}{\ell ^4(\ell +1)^4} (P_\ell '(\cos \theta ))^4\sin \theta \,\hbox {d}\theta = O\left( \frac{1}{\ell ^2}\right) . \end{aligned}$$
(3.13)

Proof

In order to investigate the first seven integrals on the l.h.s. of (3.13), we need to work with integrals of the type (cf. Lemma 3.5)

$$\begin{aligned} \left( \sqrt{\frac{2}{\pi }}\right) ^{4} \frac{2^{ \frac{c}{2} + d}}{L^{3}} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\sin \psi /L}} \right) ^{2} \sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)\,\hbox {d}\psi , \end{aligned}$$

where \(a,c,d, \in \lbrace 0,1,2 \rbrace \) and \(a+c+d =4\), a and d have same parity (0 is even). We have

$$\begin{aligned}&\left( \sqrt{\frac{2}{\pi }}\right) ^{4} \frac{2^{ \frac{c}{2} + d}}{L^{3}} \int _C^{L\pi /2} \left( \frac{1}{\sqrt{\sin \psi /L}} \right) ^{2} \sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)\,\hbox {d}\psi \nonumber \\&\quad = \frac{4}{\pi ^2} \frac{2^{ \frac{c}{2} + d}}{L^{2}} \int _C^{L\pi /2}\frac{\sin ^{a+d}(\psi + \pi /4) \sin ^{c} (\psi - \pi /4)}{\psi }\,\hbox {d}\psi + O\left( \frac{1}{L^2}\right) .\nonumber \\ \end{aligned}$$
(3.14)

There are three possibilities:

$$\begin{aligned}&\int _C^{L\pi /2}\frac{\sin ^{4}(\psi + \pi /4)}{\psi }\,\hbox {d}\psi = \int _C^{L\pi /2} \frac{ 4 \sin 2 \psi - \cos 4 \psi + 3}{8\psi }\,\hbox {d}\psi = \frac{3}{8} \log \ell + O(1)\nonumber \\&\quad \int _C^{L\pi /2}\frac{\sin ^{2}(\psi + \pi /4)\sin ^2(\psi - \pi /4)}{\psi }\,\hbox {d}\psi = \int _C^{L\pi /2} \frac{\cos 4\psi + 1}{8\psi }\,v = \frac{1}{8} \log \ell + O(1)\nonumber \\&\quad \int _C^{L\pi /2}\frac{\sin ^4(\psi - \pi /4)}{\psi }\,\hbox {d}\psi = \int _C^{L\pi /2} \frac{-4 \sin 2\psi - \cos 4\psi + 3}{8\psi }\,\hbox {d}\psi = \frac{3}{8} \log \ell + O(1).\nonumber \\ \end{aligned}$$
(3.15)

Plugging (3.15) into (3.14), we prove the first seven equality in (3.13). Concerning the last four integrals, it is simpler to deal with them and the argument is basically the same as the one used in the proof of the last equality in (3.11).

\(\square \)

3.2 Proofs of Lemmas 3.2 and 3.3

Proof of Lemma 3.2

From (3.1) with \(q=3\), we have

$$\begin{aligned} \begin{aligned} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};3) \Psi _\ell (y(\theta );3) \right] =&\frac{\ell (\ell +1)}{2}\mathbb E\Big [ \Big ( \frac{\alpha _{0,0}\beta _{3}(u)}{3!} H_{3}(f_{\ell }(\overline{x})) \\&+ \frac{\alpha _{2,0}\beta _{1}(u)}{2!} H_1(f_\ell (\overline{x}))(H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(\overline{x})) + H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(\overline{x}))) \Big ) \\&\times \Big ( \frac{\alpha _{0,0}\beta _{3}(u)}{3!} H_{3}(f_{\ell }(y(\theta ))) \\&+ \frac{\alpha _{2,0}\beta _{1}(u)}{2!} H_1(f_\ell (y(\theta )))(H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(y(\theta ))) + H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(y(\theta )))) \Big )\Big ]. \end{aligned} \end{aligned}$$

A standard application of the Diagram Formula [13, Proposition 4.15] and [19, (A1)–(A6)]

$$\begin{aligned}&{\mathbb { E}}\left[ \Psi _\ell (\overline{x};3) \Psi _\ell (y(\theta );3) \right] \nonumber \\&\quad = \frac{\ell (\ell +1)}{2} \Big ( a_1 (u) P_\ell (\cos \theta )^3 + a_2 (u) P_\ell (\cos \theta ) \frac{2}{\ell (\ell +1)}(P'_\ell (\cos \theta ) \sin \theta )^2 \nonumber \\&\qquad + a_3 (u) \frac{4}{\ell ^2(\ell +1)^2}P_\ell (\cos \theta ) (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2 \nonumber \\&\qquad + a_4 (u)\frac{4}{\ell ^2(\ell +1)^2} (P'_\ell (\cos \theta ) \sin \theta )^2 (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta ) \nonumber \\&\qquad + a_5 (u) P_\ell (\cos \theta )\frac{4}{\ell ^2(\ell +1)^2}(P'_\ell (\cos \theta ))^2 \Big ), \end{aligned}$$
(3.16)

where \(a_i(u)\), \(i=1,2, \dots , 5\) are constants depending on u (we do not need to compute them explicitly). It remains to prove that, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \int _{C/\ell } ^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};3) \Psi _\ell (y(\theta );3) \right] = O(1) \end{aligned}$$

that immediately follows plugging (3.11) in (3.16). \(\square \)

To prove Lemma 3.3, we need to recall the following identities (coming from (2.6) and (2.5)):

$$\begin{aligned} \alpha _{00}\beta _{4}(u)= & {} \sqrt{\frac{\pi }{2}}\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{u^{2}}{2}\right) (u^{4}-6u^{2}+3)=\frac{1}{2}\exp \left( -\frac{u^{2}}{2} \right) (u^{4}-6u^{2}+3 ), \nonumber \\ \alpha _{20}\beta _{2}(u)= & {} \frac{1}{2}\sqrt{\frac{\pi }{2}}\frac{1}{\sqrt{ 2\pi }}\exp \left( -\frac{u^{2}}{2}\right) (u^{2}-1 )=\frac{1}{4}\exp \left( -\frac{u^{2}}{2} \right) (u^{2}-1), \nonumber \\ \alpha _{40}\beta _{0}(u)= & {} -\frac{3}{8}\sqrt{\frac{\pi }{2}}\frac{1}{\sqrt{ 2\pi }}\exp \left( -\frac{u^{2}}{2}\right) =-\frac{3}{16}\exp \left( -\frac{u^{2}}{2}\right) . \end{aligned}$$
(3.17)

Proof of Lemma 3.3

From (3.1) with \(q=4\), we have

$$\begin{aligned} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};4) \Psi _\ell (y(\theta );4) \right]&= \frac{\ell (\ell +1)}{2}\mathbb E\Big [ \Big ( \frac{\alpha _{0,0}\beta _{4}(u)}{4!} H_{4}(f_{\ell }(\overline{x})) \nonumber \\&+ \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!} H_2(f_\ell (\overline{x}))(H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(\overline{x})) + H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(\overline{x}))) \nonumber \\&+ \frac{\alpha _{2,2}\beta _{0}(u)}{2! 2!} H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(\overline{x})) H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(\overline{x})) \nonumber \\&+ \frac{\beta _0(u) \alpha _{4,0}}{4!} ( H_4(\widetilde{\partial }_{1;x}f_{\ell }(\overline{x})) + H_4(\widetilde{\partial }_{2;x}f_{\ell }(\overline{x})) \Big ) \nonumber \\&\times \Big ( \frac{\alpha _{0,0}\beta _{4}(u)}{4!} H_{4}(f_{\ell }(y(\theta ))) \nonumber \\&+ \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!} H_2(f_\ell (y(\theta )))(H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(y(\theta ))) + H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(y(\theta )))) \nonumber \\&+ \frac{\alpha _{2,2}\beta _{0}(u)}{2! 2!} H_{2}(\widetilde{\partial }_{1;x}f_{\ell }(\overline{y}(\theta ))) H_{2}(\widetilde{\partial }_{2;x}f_{\ell }(\overline{y}(\theta )))\nonumber \\&+ \frac{\beta _0(u) \alpha _{4,0}}{4!} ( H_4(\widetilde{\partial }_{1;x}f_{\ell }(y(\theta ))) + H_4(\widetilde{\partial }_{2;x}f_{\ell }(y(\theta ))) \Big ) \Big ]. \end{aligned}$$
(3.18)

Thanks again to Diagram Formula [13, Proposition 4.15], [19, (A1)–(A6)], and, in particular, recalling [28, (108), Appendix A], we get

$$\begin{aligned}&\mathbb E\left[ \Psi _\ell (\overline{x};4) \Psi _\ell (y(\theta );4) \right] = \frac{\ell (\ell +1)}{2} \Big [ \left( \frac{\alpha _{0,0}\beta _{4}(u)}{4!}\right) ^2 4! P_\ell (\cos \theta )^4 \nonumber \\&\quad + 2\cdot \frac{\alpha _{0,0}\beta _{4}(u)}{4!} \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!} 4! P_\ell ^2(\cos \theta ) \frac{2}{\ell (\ell +1)} (P'_\ell (\cos \theta ) \sin \theta )^2 \nonumber \\&\quad + \left( \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!}\right) ^2 \Big ( 4 P_\ell ^2(\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2 \nonumber \\&\quad - 16 P_\ell (\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} (P'_\ell (\cos \theta ) \sin \theta )^2 (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta ) \nonumber \\&\quad + 4 \frac{4}{\ell ^2(\ell +1)^2}(P'_\ell (\cos \theta )\sin \theta )^4 \Big ) \nonumber \\&\quad + 2\cdot \frac{\beta _0(u)\alpha _{40}}{4!} \frac{\beta _4(u)\alpha _{00}}{4!} 4! \frac{4}{\ell ^2(\ell +1)^2}(P'_\ell (\cos \theta )^4 \sin ^4 \theta ) \nonumber \\&\quad + 2 \cdot \frac{\beta _0(u)\alpha _{40}}{4!} \frac{\beta _2(u) \alpha _{20}}{2! 2!} 4! \frac{8}{\ell ^3(\ell +1)^3}(P'_\ell (\cos \theta )\sin \theta )^2 \nonumber \\&\quad \times (P_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2\nonumber \\&\quad + \left( \frac{\beta _0(u)\alpha _{40}}{4!} \right) ^2 4! \frac{2^4}{\ell ^4(\ell +1)^4} (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^4 \nonumber \\&\quad + \left( \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!}\right) ^2 4 \cdot P_\ell ^2(\cos \theta ) \frac{4}{\ell ^2(\ell +1)^2} P'_\ell (\cos \theta )^2 \nonumber \\&\quad + 2 \cdot \frac{\alpha _{2,0}\beta _{2}(u)}{2! 2!} \frac{\alpha _{2,2}\beta _{0}(u)}{2! 2!} 4 \cdot \frac{2}{\ell (\ell +1)} (P'_\ell (\cos \theta ) \sin \theta )^2 \frac{4}{\ell ^2 (\ell +1)^2} P'_\ell (\cos \theta )^2 \nonumber \\&\quad + \left( \frac{\alpha _{2,2}\beta _{0}(u)}{2! 2!} \right) ^2 4 \cdot \frac{4}{\ell ^2(\ell +1)^2}(P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^2 \nonumber \\&\quad \times \frac{4}{\ell ^2 (\ell +1)^2} P'_\ell (\cos \theta )^2 + \left( \frac{\beta _0(u)\alpha _{40}}{4!} \right) ^2 4! \frac{2^4}{\ell ^4(\ell +1)^4} (P_\ell '(\cos \theta ))^4 \Big ]. \end{aligned}$$
(3.19)

Plugging (3.13) and (3.17) into (3.19) we get, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} 2 \cdot 8\pi ^2 \int _{C/\ell } ^{\pi /2} \mathbb E\left[ \Psi _\ell (\overline{x};4) \Psi _\ell (y(\theta );4) \right] \sin \theta \,\hbox {d}\theta = \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell + O(1) \end{aligned}$$

that allows to conclude the proof. \(\square \)

3.3 Proof of Lemma 3.4

This part is inspired by the proof of Lemma 3.5 in [8]. We will need the following estimate that is easy to check once recalling (3.10). As \(\ell \rightarrow +\infty \)

$$\begin{aligned}&\max _{\begin{array}{c} a,b,c,d \ge 0\\ a+b+c+d =5 \end{array}} \left( \frac{2}{\ell (\ell +1)} \right) ^{\frac{1}{2} b +c+d} \int _{C/\ell }^{\pi /2} |P_\ell (\cos \theta )|^{a} |P'_\ell (\cos \theta )|^{b} \nonumber \\&\quad \times |P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta |^{c} |P'_\ell (\cos \theta ) \sin \theta |^{d}\sin \theta \,\hbox {d}\theta \nonumber \\&\quad = O\left( \frac{1}{L^2} \int _C^{L\pi /2} \psi ^{-3/2 - b}\, \hbox {d}\psi \right) = O\left( \frac{1}{\ell ^2}\right) . \end{aligned}$$
(3.20)

Proof of Lemma 3.4

This proof is similar to that of Lemma 3.1 in Appendix, for further details see the proof therein. For any \(C>0\),

$$\begin{aligned}&\left| \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \,\hbox {d}\theta \right| \nonumber \\&\quad = \Big | \frac{\ell (\ell +1)}{2}\sum _{2a + 2b + c = q}\sum _{2a' + 2b' + c'=q} \frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!}\frac{\alpha _{2a',2b'}\beta _{c'}(u)}{(2a')!(2b')!(c')!} \nonumber \\&\qquad \times \frac{1}{L}\int _{C/\ell }^{\pi /2} \mathbb E\Big [H_{c}(f_{\ell }(\overline{x}))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(\overline{x}))H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(\overline{x})) \nonumber \\&\qquad \times H_{c'}(f_{\ell }(y(\theta )))H_{2a}(\widetilde{\partial }_{1;x}f_{\ell }(y(\theta ))) H_{2b}(\widetilde{\partial }_{2;x}f_{\ell }(y(\theta ))) \Big ] \sin \theta \,\hbox {d}\theta \Big | \nonumber \\&\quad \le \ell (\ell +1) \sum _{2a + 2b + c = q}\sum _{2a' + 2b' + c'=q} \left| \frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!} \right| \left| \frac{\alpha _{2a',2b'}\beta _{c'}(u)}{(2a')!(2b')!(c')!} \right| \nonumber \\&\qquad \times \Big | \mathcal V(2a,2b,c,2a',2b',c') \Big |, \end{aligned}$$
(3.21)

\(\mathcal V(2a,2b,c,2a',2b',c') \) being the sum of no more than q! terms of the form

$$\begin{aligned}&\left( \frac{2}{\ell (\ell +1)} \right) ^{\frac{1}{2} m_2 +m_3 +m_4}\int _{C/\ell }^{\pi /2} P_\ell (\cos \theta )^{m_1} (P'_\ell (\cos \theta )\sin \theta )^{m_2} \nonumber \\&\qquad \times (P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta )^{m_3} P'_\ell (\cos \theta )^{m_4}\sin \theta \,\hbox {d}\theta , \end{aligned}$$
(3.22)

where \(m_1, \dots , m_4 \ge 0\) and \(m_1+m_2+m_3+m_4 = q\). Now for some \(0<\delta <1\) and C large enough (see the proof of Lemma 3.1 in Appendix),

$$\begin{aligned}&\left| \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \,\hbox {d}\theta \right| \\&\quad \le \ell (\ell +1) \sum _{2a + 2b + c = q}\sum _{2a' + 2b' + c'=q} \left| \frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!} \right| \left| \frac{\alpha _{2a',2b'}\beta _{c'}(u)}{(2a')!(2b')!(c')!} \right| \\&\qquad \times q! (1-\delta )^{q-5} \max _{m_1+\dots + m_4 =5} \left( \frac{2}{\ell (\ell +1)} \right) ^{\frac{1}{2} m_2 +m_3 +m_4}\int _{C/\ell }^{\pi /2} |P_\ell (\cos \theta )|^{m_1} \\&\qquad \times |P'_\ell (\cos \theta )\sin \theta |^{m_2} |P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta |^{m_3} |P'_\ell (\cos \theta )|^{m_4}\sin \theta \,\hbox {d}\theta . \end{aligned}$$

Therefore, we have

$$\begin{aligned}&\left| \sum _{q=5}^{+\infty } \int _{C/\ell }^{\pi /2} {\mathbb { E}}\left[ \Psi _\ell (\overline{x};q) \Psi _\ell (y(\theta );q) \right] \sin \theta \,\hbox {d}\theta \right| \nonumber \\&\quad \le \ell (\ell +1) (1-\delta )^{-5} \max _{m_1+\dots + m_4 =5} \left( \frac{2}{\ell (\ell +1)} \right) ^{\frac{1}{2} m_2 +m_3 +m_4}\frac{1}{L}\int _C^{L\pi /2} |P_\ell (\cos \theta )|^{m_1} \nonumber \\&\qquad \times |P'_\ell (\cos \theta )\sin \theta |^{m_2} |P'_\ell (\cos \theta )\cos \theta - P_\ell ^{''}(\cos \theta )\sin ^2\theta |^{m_3} |P'_\ell (\cos \theta )|^{m_4}\sin \theta \,\hbox {d}\theta \nonumber \\&\qquad \times \sum _{q=5}^{+\infty } q! (1-\delta )^q \sum _{2a + 2b + c = q}\sum _{2a' + 2b' + c'=q} \left| \frac{\alpha _{2a,2b}\beta _{c}(u)}{(2a)!(2b)!c!} \right| \left| \frac{\alpha _{2a',2b'}\beta _{c'}(u)}{(2a')!(2b')!(c')!} \right| . \end{aligned}$$
(3.23)

The series on the r.h.s. of (3.23) being convergent, (3.20) applied in (3.23) allows to conclude the proof. \(\square \)

4 Proof of Proposition 2.4

Recall that \( L:= \ell + \frac{1}{2}, \) and set

$$\begin{aligned} \mathcal J_\ell (\psi ;4) := - \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\frac{1}{4!} \mathcal {G}(u) \frac{8\pi ^{2}}{L}{\mathbb { E}}\left[ \Psi _\ell (\overline{x};4) H_{4}(f_{\ell }(y(\psi /L)))\right] . \end{aligned}$$

We will need the following key result (inspired by Proposition 3.1 in [19]) whose proof is in Appendix.

Lemma 4.1

For any \(C>0\), uniformly over \(\ell \) we have, for \(0<\psi <C\),

$$\begin{aligned} \mathcal J_\ell (\psi ;4) = O(\ell ), \end{aligned}$$
(4.1)

and, for \(C<\psi < L\frac{\pi }{2}\),

$$\begin{aligned} \mathcal J_\ell (\psi ;4)= & {} \frac{1}{4} \frac{\pi }{2}\phi (u)^2 \left( H_4(u) + 2H_2(u) -\frac{3}{8} \right) ^2 \frac{1}{\psi \sin \psi /L} \nonumber \\&+ a(u) \frac{\cos 4\psi }{\psi \sin \psi /L} + b(u) \frac{\sin 2\psi }{\psi \sin \psi /L} \nonumber \\&+O\left( \frac{1}{\psi ^2 \sin ^{2}\frac{\psi }{L}}\right) , \end{aligned}$$
(4.2)

where a(u) and b(u) are two (explicit) constants that depend on u.

Lemma 4.2

We have

$$\begin{aligned} \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text { proj}[\mathcal L_\ell (u)|q],\mathcal {M}_{\ell }(u)\right) =\int _{0}^{L\pi } \mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\hbox {d}\psi . \end{aligned}$$

Proof

By continuity of the inner product in \(L^{2}\) spaces, we have

$$\begin{aligned} \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text { proj}[\mathcal L_\ell (u)|q], \mathcal {M}_{\ell }(u)\right) = \lim _{\varepsilon \rightarrow 0}\mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L^\varepsilon _\ell (u)|q], \mathcal {M}_{\ell }(u)\right) . \end{aligned}$$

Recall (3.4). Note that \( \sum _{q=3}^{+\infty } \Psi ^\varepsilon _\ell (\cdot ;q)\) and \(H_{4}(f_{\ell }(\cdot ))\) are both in \( L^{2}(\mathbb {S}^{2}\times \Omega )\) and they are isotropic (as we will see just below, we need isotropy to pass from a double integral over the sphere to an integral over the geodesic meridian), and thus,

$$\begin{aligned}&\mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L^\varepsilon _\ell (u)|q], \mathcal {M}_{\ell }(u)\right) = \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \int _{\mathbb S^2} \Psi ^\varepsilon _\ell (x;q)\,\hbox {d}x,\mathcal {M}_{\ell }(u)\right) \\&\quad = - \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!}{\mathbb { E}}\left[ \int _{ \mathbb {S}^{2}}\sum _{q=3}^{+\infty } \Psi ^\varepsilon _\ell (x;q)\,\hbox {d}x \int _{\mathbb {S}^{2}}H_{4}(f_{\ell }(y))\hbox {d}y\right] \\&\quad = - \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!}\int _{ \mathbb {S}^{2}} \int _{\mathbb {S}^{2}}{\mathbb { E}}\left[ \sum _{q=3}^{+\infty } \Psi ^\varepsilon _\ell (x;q) H_{4}(f_{\ell }(y))\right] \,\hbox {d}x \hbox {d}y \\&\quad = - \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!} \times 8\pi ^{2}\int _{0}^{\pi }{\mathbb { E}}\left[ \sum _{q=3}^{+\infty } \Psi ^\varepsilon _\ell (\overline{x};q) H_{4}(f_{\ell }(y(\theta )))\right] \sin \theta \, \hbox {d}\theta \\&\quad =- \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!} \times 8\pi ^{2}\int _{0}^{\pi }{\mathbb { E}}\left[ \Psi ^\varepsilon _\ell (\overline{x};4) H_{4}(f_{\ell }(y(\theta )))\right] \sin \theta \, \hbox {d}\theta . \end{aligned}$$

The integrand \({\mathbb { E}}\left[ \Psi ^\varepsilon _\ell (\overline{x};4) H_{4}(f_{\ell }(y(\theta )))\right] \) can be computed explicitly, and it is easily seen to be absolutely bounded for fixed \(\ell \), uniformly over \( \varepsilon ,\) see Lemma 4.1. Hence, by Lebesgue theorem we may exchange the limit and the integral, and we have that

$$\begin{aligned}&\mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u)|q], \mathcal {M}_{\ell }(u)\right) =\lim _{\varepsilon \rightarrow 0}\mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell ^\varepsilon (u)|q], \mathcal {M}_{\ell }(u)\right) \\&\quad =- \sqrt{ \frac{\lambda _{\ell }}{2}}\sqrt{\frac{\pi }{2}}\mathcal {G}(u) \frac{1}{4!} \times 8\pi ^{2}\int _{0}^{\pi }{\mathbb { E}}\left[ \Psi _\ell (\overline{x};4) H_{4}(f_{\ell }(y(\theta )))\right] \sin \theta \, \hbox {d}\theta . \end{aligned}$$

Performing the change of variable \(\psi = L\theta \), we can now write

$$\begin{aligned} \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u)|q],\mathcal {M}_{\ell }(u)\right) =\int _{0}^{L\pi } \mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\hbox {d}\psi . \end{aligned}$$

\(\square \)

Proof of Proposition 2.4

It suffices to prove that, as \(\ell \rightarrow +\infty \),

$$\begin{aligned} \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u)|q], \mathcal {M}_{\ell }(u)\right) = \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell + O(1) \end{aligned}$$
(4.3)

(cf. (2.13) and Proposition 2.3). From Lemma 4.2 we can write, for any \(C>0\),

$$\begin{aligned} \mathrm{Cov\,}\left( \sum _{q=3}^{+\infty } \text {proj}[\mathcal L_\ell (u)|q],\mathcal {M}_{\ell }\right) =\int _{0}^{C}\mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\hbox {d}\psi +2\int _{C}^{L\pi /2}\mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\hbox {d}\psi .\nonumber \\ \end{aligned}$$
(4.4)

For the first summand in (4.4), we have, thanks to (4.1),

$$\begin{aligned} \int _{0}^{C}\mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\,\hbox {d}\psi = O \left( \ell \int _{0}^{C}\sin \frac{\psi }{L}\,\hbox {d}\psi \right) = O\left( \frac{\ell }{L}\int _{0}^{C}\psi \, \hbox {d}\psi \right) =O(1)\text { , as }\ell \rightarrow \infty ; \end{aligned}$$

for the second summand in (4.4), using Lemma 4.1 (precisely (4.2)) and integrating, we obtain

$$\begin{aligned} 2\int _{C}^{L\pi /2}\mathcal {J}_{\ell }(\psi ;4)\sin \frac{\psi }{L}\,\hbox {d}\psi = \frac{\pi }{4} \mathcal {G}(u)^2 \log \ell +O(1) \end{aligned}$$

giving (4.3). \(\square \)