The defect variance of random spherical harmonics

In recent years, a lot of interest has been drawn by the analysis of the geometric features of random eigenfunctions for the spherical Laplacian. More precisely, let us denote as usual by {Y_lm( · )}_{m = −l, ..., l}, for l = 1, 2, ..., the set of (real-valued) spherical harmonics, i.e. the array of real-valued functions satisfying

$$\begin{equation*} \Delta _{S^{2}}Y_{lm}=-l(l+1)Y_{lm}, \end{equation*}$$

where $\Delta _{S^{2}}$ denotes the spherical Laplacian (see for instance [16, 18]). The random model we shall focus on is

$$\begin{equation} f_{l}(x)=\frac{1}{\sqrt{2l+1}}\sum _{m=-l}^{l}a_{lm}Y_{lm}(x), \end{equation} \tag{ 1 }$$

where the coefficients {a_lm} are independent standard Gaussian with zero-mean and unit variance.

The random field f_l is isotropic, meaning that for any $k\in \mathbb {N}$ and $x_{1},\ldots ,x_{k}\in \mathcal {S}^{2}$, the distribution of the random vector {f_l(x₁), ..., f_l(x_l)} is invariant under rotations, i.e. equal to the distribution of {f_l(g · x₁), ..., f_l(g · x_l)} for any rotation g ∈ SO(3) on the sphere. Also, f_l is centred Gaussian, with the covariance function

$$\begin{equation} r_{l}(x,y):=\mathbb {E}[f_{l}(x)\cdot f_{l}(y)]=P_{l}(\cos (d(x,y))), \end{equation} \tag{ 2 }$$

where P_l are the usual Legendre polynomials defined by Rodrigues' formula

$$\begin{equation*} P_{l}(t):=\frac{1}{2^{l}l!}\frac{{\rm d}^{l}}{{\rm d}t^{l}}(t^{2}-1)^{l} \end{equation*}$$

and d(x, y) is the (spherical) geodesic distance between x and y. As is well known, Legendre polynomials are orthogonal w.r.t. the constant weight ω(t) ≡ 1 on [ − 1, 1], see section 5 for more discussion and details.

1.1. Background

Random eigenfunctions for the spherical Laplacian naturally emerge in a number of different physical contexts. A particularly active area is related to the analysis of isotropic spherical random fields on the sphere, as motivated for instance by the analysis of cosmic microwave background radiation (CMB), see for instance [4, 5] and the references therein. Under these circumstances, eigenfunctions like f_l represent the (normalized) Fourier components of the field, i.e. the following orthogonal expansion holds, in the L² sense:

$$\begin{equation*} f(x)=\sum _{l=1}^{\infty }c_{l}f_{l}(x), \end{equation*}$$

where c_l ⩾ 0 is a deterministic sequence (depending on the angular power spectrum) which encodes the full correlation structure of f.

In the CMB literature, f(x), and hence the components f_l, are actually observed by highly sophisticated satellite experiments such as WMAP by NASA and Planck by ESA. It is then common practice to analyse geometric functionals of the observed CMB radiation to constraint the statistical properties of the underlying fields, e.g. to test for isotropy and/or Gaussianity. For instance, the three Minkowski functionals, providing the area, the boundary length and the topological genus of excursion sets over a given level $( A_{z}:=\lbrace x\in \mathcal {S} ^{2}:f(x)\ge z\rbrace )$ have been applied on CMB data by a huge number of authors, including [7, 8, 12]. Much effort has been spent in analysing unexpected features, for instance, the so-called cold spot (see [2, 3] and the references therein), whose statistical significance can also be evaluated by means of local curvature properties of isotropic Gaussian fields [6].

In this work, we shall focus on one of the most important geometric functionals, namely the defect. The defect (or 'signed area', see [1]) of a function $\psi :\mathbb {S}^{2}\rightarrow \mathbb {R}$ is defined as

$$\begin{equation} \fl \mathcal {D}(\psi ):=\mathrm{meas}( \psi ^{-1}(0,\infty )) - \mathrm{meas}( \psi ^{-1}(-\infty ,0)) =\int _{\mathcal {S}^{2}} \mathcal {H}(\psi (x))\,{\rm d}x. \end{equation} \tag{ 3 }$$

Here $\mathcal {H}(t)$ is the Heaviside function,

$$\begin{equation} \mathcal {H}(t)={\openone} _{[0,\infty ) }(t)-{\openone} _{(-\infty ,0]}(t)= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1 & t>0 \\ -1 & t<0 \\ 0 & t=0 \end{array}\right. , \end{equation} \tag{ 4 }$$

where ${\openone} _{A}(t)$ is the usual indicator function of the set A and dx is the Lebesgue measure. The defect is hence the difference between the areas of positive and negative inverse image of ψ, respectively.

The primary focus of this paper is the distribution of the defect $\mathcal {D}_{l}=\mathcal {D}(f_{l})$ of the random Gaussian fields f_l. Note that for odd l, f_l is always odd, so that in this case the defect vanishes identically ($\mathcal {D}_{l}\equiv 0$), and therefore $\mathcal {D}_{l}$ has the non-trivial distribution for even l only. A possible alternative to avoid trivialities is to restrict ourselves to a subset of the sphere, the most natural choice being a hemisphere, i.e. we may choose any hemisphere $\mathcal {E}\subseteq \mathcal {S}^{2}$ and define

$$\begin{equation*} \mathcal {D}_{l}^{\mathcal {E}}:=\int _{\mathcal {E}}\mathcal {H}(f_{l}(x))\,{\rm d}x; \end{equation*}$$

in this paper we study $\mathcal {D}_{l}$ for even l only.

1.2. Statement of the main result

Evaluating the expectation of $\mathcal {D}_{l}$ is trivial. Indeed, note that integration over $\mathcal {S}^{2}$ is exchangeable with expectation, so that

$$\begin{equation*} \mathbb {E}[\mathcal {D}_{l}]=\int _{\mathcal {S}^{2}}\mathbb {E}\left[ \mathcal {H }(f_{l}(x))\right] {\rm d}x, \end{equation*}$$

and $\mathbb {E}\left[ \mathcal {H}(f_{l}(x))\right] =0$ vanishes for every $x\in \mathcal {S}^{2}$, by the symmetry of the Gaussian distribution. We just established the following lemma.

Lemma 1.1. For every l = 1, 2, ..., we have

$$\begin{equation*} \mathbb {E}[\mathcal {D}_{l}]=0. \end{equation*}$$

The main result of this paper concerns the asymptotic behaviour of the defect variance:

Theorem 1.2. As l → ∞ along even integers, the defect variance is asymptotic to

$$\begin{equation} {\rm Var}(\mathcal {D}_{l})=\frac{C}{l^{2}}(1+o(1)), \end{equation} \tag{ 5 }$$

where C > 0 is a positive constant.

The constant C in (5) may be expressed in terms of the infinite (conditionally convergent) integral

$$\begin{equation} C=32\pi \int \nolimits _{0}^{\infty }\psi ( \arcsin (J_{0}(\psi ))-J_{0}(\psi ))\, {\rm d}\psi . \end{equation} \tag{ 6 }$$

See section 2 for some details on the integral on the RHS of (6). We do not know whether one can evaluate C explicitly; however, we shall be able to show that

$$\begin{equation*} C>\case{32}{\sqrt{27}} \end{equation*}$$

(see lemma 5.5).

One should compare the statement of theorem 1.2 to empirical results of the study conducted by Blum, Gnutzmann and Smilansky [1]. The authors of that work studied the defect (or, as they refer to it, the 'signed area') of random monochromatic waves on various planar domains. For the particular case of a unit circle, they found that the order of magnitude of defect variance is consistent to (5) per unit area, with leading constant evaluated numerically as ≈0.0386. The corresponding defect variance per unit area in our situation is

$$\begin{equation*} {\rm Var}\left(\frac{\mathcal {D}_{l}}{4\pi } \right)=\frac{\widetilde{C}}{l^{2}} (1+o(1)) \end{equation*}$$

with the leading constant³

$$\widetilde{C}>\frac{2}{\pi ^{2}\times \sqrt{27}} = 0.0389\ldots .$$

1.3. Previous work

To put our results in a proper perspective and explain the technical difficulties to be handled, we need to briefly recall some results from [11]. In that work we studied the asymptotic behaviour of excursion sets

$$A_{z} = f_{l}^{-1}\left([z,\infty )\right)$$

of f_l, as l → ∞. As we will recall below, a rather peculiar phenomenon can be shown to hold for z ≠ 0; namely, the asymptotic distribution of the area of these excursion sets is fully degenerate over z, i.e. it corresponds to a Gaussian random variable times a deterministic function of z. This result is intuitively due to an asymptotic degeneracy in the excursion set functionals, which turns out to be dominated by a single polynomial (quadratic) term.

This degeneracy, however, does not hold for the special case z = 0. More precisely, in [11] we focused on the asymptotic behaviour of the empirical measure of random spherical harmonics, defined as

$$\begin{equation*} \Phi _{l}(z):=\int _{S^{2}}{\openone} _{(-\infty ,z]}(f_{l}(x))\,{\rm d}x=\mathrm{meas} \left\lbrace x:f_{l}(x)\le z\right\rbrace , \end{equation*}$$

$z\in \mathbb {R}$. A key step in that paper is the asymptotic expansion

$$\begin{equation} \Phi _{l}(z)=4\pi \times \Phi (z)+\sum _{q=1}^{\infty }\frac{V_{q}(z)}{q!} h_{l;q},h_{l;q}:=\int _{\mathbb {S}^{2}}H_{q}(f_{l}(x))\,{\rm d}x, \end{equation} \tag{ 7 }$$

where H_q(.) are standard Hermite polynomials, $\Phi (z)=\Pr \left\lbrace Z\le z\right\rbrace$ is the cumulative distribution function of a standard Gaussian variable and the deterministic functions V_q(z) can be explicitly provided in terms of higher order derivatives of Φ(z), V_q(z) = ( − 1)^qΦ^(q)(z) (see [11] for more discussion and details).

It turns out that, as l → ∞,

$$\begin{equation*} \frac{\sqrt{l}}{4\pi }h_{l;2}\mathop{\rightarrow}_{d} N(0,1) \end{equation*}$$

converges by distribution to the standard Gaussian, whereas for q ⩾ 3,

$$\begin{equation*} \sqrt{l}h_{l;q}\mathop{\rightarrow}_{d} 0; \end{equation*}$$

moreover, V₂(z) = −zϕ(z), where ϕ(z) is the standard Gaussian probability density, clearly does not vanish for all z ≠ 0, whence the asymptotic behaviour of

$$\sqrt{l}\left\lbrace \Phi _{l}(z)-4\pi \times \Phi (z)\right\rbrace$$

is easily seen to be Gaussian, uniformly over z. Furthermore, the limiting process is completely degenerate with respect to z, a feature to which we shall come back later.

For z ≠ 0, the asymptotic behaviour of the area functional for the excursion sets is hence fully understood. The previous argument, however, fails for z = 0, as in this case the leading term is null and each summand in the asymptotic expansion (7) becomes relevant. Up to a linear transformation, this is clearly equivalent to the defect functional, indeed

$$\begin{equation*} \mathcal {D}_{l}=4\pi -2\Phi _{l}(0). \end{equation*}$$

Thus, the case z = 0 is the most challenging from the mathematical point of view, and, at the same time, the most interesting from the point of view of geometric interpretation.

1.4. Overview of the paper

To establish our results, we shall need a detailed analysis of the odd moments of Legendre polynomials

$$\begin{equation*} \int \nolimits _{0}^{1}P_{l}(t )^{2k+1}\,{\rm d}t = \int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )^{2k+1}\sin {\theta }\,{\rm d}\theta . \end{equation*}$$

The rationale for this can be explained as follows. It is relatively easy to express (up to a constant) the defect variance as

$$\begin{equation} I_{l}=\int \nolimits _{0}^{\pi /2}\arcsin (P_{l}(\cos \theta ))\sin {\theta }\, {\rm d}\theta , \end{equation} \tag{ 8 }$$

where sin θdθ is (up to a constant) the uniform measure on the sphere in the spherical coordinates (see lemma 4.1). It then remains to understand the asymptotic behaviour of I_l, a task which is put forward in proposition 4.2; here we provide the main ideas underlying its proof. We know from Hilb's asymptotics (lemma 5.1) that P_l(cos θ) has a scaling limit: as l → ∞, for any fixed

$$\begin{equation*} \psi \in \left[ 0,(l+1/2)\frac{\pi }{2}\right] , \end{equation*}$$

we have

$$\begin{equation} P_{l}\left( \cos \left( \frac{\psi }{l+1/2}\right) \right) \approx J_{0}(\psi ). \end{equation} \tag{ 9 }$$

In fact, the latter estimate holds uniformly for ψ = o(l), and it may be shown that, as a tail of a conditionally convergent integral, the contribution of the other regime ψ ≳ l is negligible (see the proof of proposition 4.2).

In the following, we neglect the difference between l and $l+\frac{1}{2}$. It is then natural to try to replace P_l(cos θ) in (8) with its scaling limit; a formal substitution yields heuristically

$$\begin{eqnarray} &&\fl I_{l} \approx \frac{1}{l}\int \nolimits _{0}^{l\pi /2}\arcsin ( P_{l}( \cos ( \psi /l) ) ) \sin ( \psi /l)\, {\rm d}\psi \approx \frac{1}{l}\int \nolimits _{0}^{l\pi /2}\arcsin (J_{0}(\psi ))\sin ( \psi /l)\, {\rm d}\psi \nonumber\\ &&\hphantom{\fl I_{l} }\approx \frac{1}{l^{2}}\int \nolimits _{0}^{l\pi /2}\arcsin (J_{0}(\psi ))\psi\, {\rm d}\psi , \end{eqnarray} \tag{ 10 }$$

where we replaced $\sin \big( \frac{\psi }{l}\big)$ with $\frac{\psi }{l}$, which is justified for ψ = o(l). This is inconsistent with the statement of theorem 1.2; the integral

$$\begin{equation} \int \nolimits _{0}^{\infty }\arcsin (J_{0}(\psi ))\psi\, {\rm d}\psi , \end{equation} \tag{ 11 }$$

diverges, so some more care is needed to transform (10). Moreover, the last integrand in (10) is somewhat different from the integrand in integral (6) defining C (neglecting the constant in front of the integral in (6)) in that we subtract J₀(ψ) from the arcsine. Note that the factor ψ in both (10) and (6) is reminiscent of the uniform measure sin θ dθ on the sphere.

There is a subtlety that explains the latter discrepancy between the integrand in definition (6) of C and the integrand in (10). One way to justify the substitution of the scaling limit of P_l(cos θ) (i.e. the second step of (10)) is expanding the arcsine in the integral of (8) into the Taylor series around the origin

$$\begin{equation*} \arcsin (t)=\sum \limits _{k=0}^{\infty }a_{k}t^{2k+1}, \end{equation*}$$

for some explicitly given coefficients a_k (see (18) and the formula immediately after). We then need to evaluate all the odd moments of P_l(cos θ) w.r.t. the measure sin θ dθ on $\left[ 0,\frac{\pi }{2}\right]$. It turns out that the moments (appropriately scaled) are asymptotically equivalent to the corresponding moments of the Bessel function on $\mathbb {R}_{+}$ w.r.t. the measure ψ dψ, with the exception for the first moment (lemma 5.2). Indeed, P_l(cos θ) integrates to zero, whereas $\int \nolimits _{0}^{\infty }J_{0}(\psi )\psi\, {\rm d}\psi$ diverges; this phenomenon also accounts for the divergence of (11). To account for this difference, we need to subtract J₀(ψ) from the arcsine in integrand (10), thus obtaining a conditionally convergent integral. This is indeed the heuristic explanation for the discrepancy between (10) and (6).

To summarize this discussion, we note the following. The variance is given, up to an explicit constant, by integral (8). The constant C in (6) is reminiscent of I_l up to scaling (9): the expression P_l(cos θ) is replaced by its scaling limit J₀(ψ), and the measure sin θdθ becomes ψdψ on $\mathbb {R}_{+}$ after scaling. Subtracting J₀(ψ) from the integrand in (6) accounts for the fact that for even l, the integral of P_l(cos θ) against sin θdθ vanishes, whereas this is not the case of integral of the scaling limit J₀(ψ) against ψdψ.

As reported in section 1.3, the asymptotics for the variance of the excursion sets is given for z ≠ 0 by

$$\begin{equation*} {\rm Var}(\Phi _{l}(z))\sim z^{2}\phi (z)^{2}\cdot \frac{4\pi ^2}{l} \end{equation*}$$

(here ϕ is the Gaussian probability density); for the defect (z = 0), this gives only an upper bound $o\left( \frac{1}{l}\right)$. The principal result of this paper (theorem 1.2) states that the defect variance (5) is of the order of magnitude l⁻². Our explanation for this discrepancy is the disappearance, for z = 0, of the quadratic term in the Hermite expansion of the function ${\openone} (f_{l}\le z),$ see [11].

We should compare this situation to the length distribution of the level curves. For $t\in \mathbb {R}$, let

$$\begin{equation*} \mathcal {L}_{l}^{t}=\mathcal {L}^{t}(f_{l})=\mathrm{length}\left( f_{l}^{-1}(t)\right) \end{equation*}$$

be the (random) length of the level curve f⁻¹_l(t), the most important case being that of t = 0; the corresponding level curve is called the nodal line. It is known [19] that for $t\in \mathbb {R}$, the expected length is

$$\begin{equation*} \mathbb {E}\left[ \mathcal {L}_{l}^{t}\right] \sim c_{1}\,{\rm e}^{-t^{2}/2}\cdot l. \end{equation*}$$

The variance is asymptotic to

$$\begin{equation*} {\rm Var}\left( \mathcal {L}_{l}^{t}\right) \sim c_{2}\,{\rm e}^{-t^{2}}t^{4}\cdot l \end{equation*}$$

for t ≠ 0, whereas for the nodal length,

$$\begin{equation*} {\rm Var}\left( \mathcal {L}_{l}^{0}\right) \sim c_{3}\log {l}. \end{equation*}$$

It is therefore natural to conjecture that the discrepancy of the defect and the empirical measure is somehow related to the discrepancy in the variance of the nodal length. However, no simple explanation, as in the case of the defect, is known to explain the discrepancy between the nodal length and the non-vanishing level curve length.

Recall that for the two random variables X, Y, the correlation is defined as

$$\begin{equation*} \mathrm{Corr}(X,Y)=\frac{{\rm Cov}(X,Y)}{\sqrt{{\rm Var}(X)}\sqrt{{\rm Var}(Y)}}. \end{equation*}$$

It is known [19] that for every $t_{1},t_{2}\in \mathbb {R}$, $\mathcal {L}_{l}^{t}$ become asymptotically fully dependent for large l, in the sense that⁴

$$\begin{equation} \mathrm{Corr}\left( \mathcal {L}_{l}^{t_{1}},\mathcal {L}_{l}^{t_{2}}\right) =1-o_{l\rightarrow \infty }(1). \end{equation} \tag{ 12 }$$

In fact, a much stronger statement regarding the rate of convergence was proven. It then seems that the asymptotic degeneracy of Φ_l(z) is related to the asymptotic full dependence of the level lengths (12).

A possible explanation for phenomenon (12) is the following conjecture due to Mikhail Sodin. Let $x\in \mathcal {S}^{2}$ and for $t\in \mathbb {R}$, let $\mathcal {L}_{l;x}^{t}=\mathcal {L}_{x}^{t}(f_{l})$ (the 'local length') be the (random) length of the unique component of f⁻¹_l(t) that contains x inside (or 0, if f_l does not cross the level t).

Conjecture 3.1 (Mikhail Sodin). The local lengths are asymptotically fully dependent in the sense that for every $x\in \mathcal {S}^{2}$ and $t_{1},t_{2}\in \mathbb {R}$,

$$\begin{equation*} \mathrm{Corr}\big( \mathcal {L}_{l;x}^{t_{1}},\mathcal {L}_{l;x}^{t_{2}} \big) =1-o_{l\rightarrow \infty }(1). \end{equation*}$$

Heuristically, conjecture 3.1 is stronger than (12), since $\mathcal {L}^{t}$ can be viewed as a summation of $\mathcal {L}_{x}^{t}$ over a set of points $x\in \mathcal {S}^{2}$. More rigorously, we should assume that f_l(x) ≠ t; it is immediate to see that this is satisfied almost surely.

The first lemma is probably already known, but we failed to locate a direct reference and thus we report it for completeness. The proof is based on standard properties of Gaussian random variables.

Lemma 4.1. For even l, we have

$$\begin{equation} {\rm Var}(\mathcal {D}_{l})=32\pi \int \nolimits _{0}^{\pi /2}\arcsin (P_{l}(\cos \theta ))\sin {\theta }\,{\rm d}\theta . \end{equation} \tag{ 13 }$$

We postpone the proof of lemma 4.1 until the end of this section. Let us denote the integral in (13)

$$\begin{equation} I_{l}=\int \nolimits _{0}^{\pi /2}\arcsin (P_{l}(\cos \theta ))\sin {\theta }\, {\rm d}\theta , \end{equation} \tag{ 14 }$$

so that evaluating the defect variance is equivalent to evaluating I_l, which is done in the following proposition (for even l).

Proposition 4.2. As l → ∞ along even numbers, we have

$$\begin{equation} I_{l}=\frac{C_{1}}{l^{2}}+o_{l\rightarrow \infty }\left( \frac{1}{l^{2}} \right) , \end{equation} \tag{ 15 }$$

where

$$\begin{equation} C_{1}=\int \nolimits _{0}^{\infty }\psi \left( \arcsin (J_{0}(\psi ))-J_{0}(\psi )\right) {\rm d}\psi . \end{equation} \tag{ 16 }$$

Moreover, the constant C₁ is strictly positive.

Proof of theorem 1.2 (assuming proposition 4.2). Formula (13) together with proposition 4.2 yields

$$\begin{equation*} {\rm Var}(\mathcal {D}_{l})=\frac{C}{l^{2}}\left( 1+o_{l\rightarrow \infty }(1)\right) . \end{equation*}$$

The positivity of the constant C = 32πC₁ follows directly from the positivity of C₁, which is claimed in proposition 4.2.

Proof of proposition 4.2. As was explained in section 2, to extract the asymptotics of I_l, we will expand the arcsine on the RHS of (14) into the Taylor series around the origin; we will encounter only the odd moments of P_l(cos θ), due to the arcsine being an odd function. As was pointed out in section 2, the function P_l(cos θ) differs from its scaling limit in that the integral of the former vanishes, whereas the integral of the latter diverges; all the other odd moments of P_l(cos θ) are asymptotic to the (properly scaled) corresponding moments of the Bessel function (see lemma 5.2). To account for this discrepancy, we subtract P_l(cos θ) from the integrand and add it back separately. To this end, we write

$$\begin{equation*} \arcsin (P_{l}(\cos \theta ))=\left( \arcsin (P_{l}(\cos \theta ))-P_{l}(\cos \theta )\right) +P_{l}(\cos \theta ), \end{equation*}$$

so that the vanishing of the integral of the latter

$$\begin{equation*} \int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )\sin {\theta }\,{\rm d}\theta =0, \end{equation*}$$

for even l, implies

$$\begin{equation} I_{l}=\int \nolimits _{0}^{\pi /2}\left( \arcsin (P_{l}(\cos \theta ))-P_{l}(\cos \theta )\right) \sin {\theta }\,{\rm d}\theta . \end{equation} \tag{ 17 }$$

The advantage of the latter representation (17) over (14) is that the only powers that will appear in the Taylor expansion of the arcsine on the right-hand side of (17) are of order ⩾3, so that the moments of P_l(cos θ) are all identical to the corresponding moments of the scaling limit. Intuitively, this means that we may replace the appearances of P_l(cos θ) in (17) by its scaling limit. The rest of the present proof is a rigorous argument that will establish the latter statement.

Let

$$\begin{equation} \arcsin (t)-t=\sum \limits _{k=1}^{\infty }a_{k}t^{2k+1}, \end{equation} \tag{ 18 }$$

where

$$\begin{equation} a_{k}=\frac{(2k)!}{4^{k}(k!)^{2}(2k+1)} \end{equation} \tag{ 19 }$$

are the Taylor coefficients of the arcsine. Note that all the terms in expansion (18) are positive, and by the Stirling formula, the coefficients are asymptotic to

$$\begin{equation} a_{k}\sim \frac{c}{k^{3/2}} \end{equation} \tag{ 20 }$$

for some c > 0, so that, in particular, the Taylor series (18) is uniformly absolutely convergent. Therefore, we may write

$$\begin{equation} I_{l}=\sum \limits _{k=1}^{\infty }a_{k}\int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )^{2k+1}\sin {\theta }\,{\rm d}\theta . \end{equation} \tag{ 21 }$$

We know from lemma 5.2, that for every k ⩾ 1,

$$\begin{equation} \int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )^{2k+1}\sin {\theta }\,{\rm d}\theta \sim \frac{c_{2k+1}}{l^{2}}, \end{equation} \tag{ 22 }$$

with c_{2k + 1} given by (32); comparing this result to (21), it is natural to expect that

$$\begin{equation} I_{l}=\frac{C_{2}}{l^{2}}+o\left( \frac{1}{l^{2}}\right) , \end{equation} \tag{ 23 }$$

where

$$\begin{equation} C_{2}=\sum \limits _{k=1}^{\infty }a_{k}c_{2k+1}. \end{equation} \tag{ 24 }$$

We are going to formally prove (23) immediately; however, first we evaluate the constant C₂ by summing up the series in (24) and validate that indeed C₂ = C₁ in (16), as claimed. We plug (32) into (24) to formally compute

$$\begin{eqnarray*} &&C_{2} = \sum \limits _{k=1}^{\infty } a_{k} \int \nolimits _{0}^{\infty } \psi J_{0}(\psi )^{2k+1} {\rm d}\psi = \int \nolimits _{0}^{\infty } \psi \cdot \left( \sum \limits _{k=1}^{\infty } a_{k} J_{0}(\psi )^{2k+1}\right) {\rm d}\psi \nonumber\\ &&\hphantom{C_{2} }= \int \nolimits _{0}^{\infty } \psi \cdot ( \arcsin (J_{0}(\psi ))-J_{0}(\psi ) )\, {\rm d}\psi = C_{1}, \end{eqnarray*}$$

where to obtain the third equality we used (18) again. To justify the exchange of the summation and integration order, we consider the finite summation

$$\begin{equation*} \sum \limits _{k=1}^{m}a_{k}\int \nolimits _{0}^{\infty } \psi J_{0}(\psi )^{2k+1}\, {\rm d}\psi , \end{equation*}$$

using (20) and (30) to bound the contribution of tails, and take the limit m → ∞.

We now turn to proving (23). To this end we expand (18) into a finite-degree Taylor polynomial while controlling the tail using lemma 5.7. Indeed, using (20) and (45), we easily obtain

$$\begin{eqnarray*} &&\fl \sum _{k=m+1}^{\infty } a_{k}\int \nolimits _{0}^{\pi /2} | P_{l}(\cos \theta )|^{2k+1} \sin {\theta }\,{\rm d}\theta \le \sum _{k=m+1}^{\infty } a_{k}\int \nolimits _{0}^{\pi /2} \left| P_{l}(\cos \theta )\right|^{5} \sin {\theta }{\rm d}\theta \\ &&\hphantom{\fl \sum _{k=m+1}^{\infty } a_{k}\int \nolimits _{0}^{\pi /2} | P_{l}(\cos \theta )|^{2k+1} \sin {\theta }\,{\rm d}\theta}\lesssim \frac{1}{l^2} \cdot \sum \limits _{k=m+1}\frac{1}{k^{3/2}} \lesssim \frac{1}{ \sqrt{m}l^{2}}, \end{eqnarray*}$$

since |P_l(t)| ⩽ 1 for every l and t ∈ [ − 1, 1], so that |P_l(t)|^k is monotone decreasing with k; the notation A '≲' B here and everywhere in this paper means that there exists a constant C > 0 (which may vary) so that A ⩽ C · B.

We then have for every m (m = m(l) to be chosen)

$$\begin{equation} I_{l} = \sum \limits _{k=1}^{m}a_{k} \int \nolimits _{0}^{\pi /2} P_{l}(\cos \theta )^{2k+1}\sin {\theta }{\rm d}\theta + O\left( \frac{1}{\sqrt{m} l^{2} } \right), \end{equation} \tag{ 25 }$$

and plugging (22) (a direct consequence of lemma 5.2) into (25) finally yields

$$\begin{equation} I_{l} = C_{2,m} \cdot \frac{1}{l^2} + o_{m}\left( \frac{1}{l^2} \right) + \frac{1}{\sqrt{m}l^{2}}, \end{equation} \tag{ 26 }$$

with

$$\begin{equation*} C_{2,m} = \sum \limits _{k=1}^{m}a_{k}c_{2k+1}. \end{equation*}$$

It is clear that (26) implies (23) (recall definition (24) of C₂ and note that as m → ∞, C_{2, m} → C₂), which concludes the proof of statement (15) of the present proposition.

It then remains to prove the positivity of the constant C₁. While its nonnegativity C₁ ⩾ 0 is clear since it is the leading constant for the integral I₁, which, up to an explicit positive constant, equals the variance of a random variable via (13), the strict positivity is less obvious. To this end, we recall that C₁ = C₂, the latter being given by (24). Note that all the Taylor coefficients a_k of arcsine are positive (see (19)). Hence, the positivity of C₂ (and thus also of C₁) follows from the second statement of lemma 5.2, which claims that c_k ⩾ 0 are nonnegative, and c₃ > 0 is explicitly given.

Proof of lemma 4.1. The result of the present lemma is an artefact of the following general fact (see e.g. [13, 14]). Let (X₁, X₂) be a two-variate centred Gaussian random variable with the covariance matrix

$$\begin{equation*} \left( {\begin{array}{@{}cc@{}} 1 & r \\ r & 1 \end{array}} \right), \end{equation*}$$

|r| ⩽ 1, and for i = 1, 2, define the random variables

$$\begin{equation*} B_{i} = \mathcal {H}(X_{i}) = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1 & X_{i} > 0 \\ -1 & X_{i} < 0 \end{array}\right., \end{equation*}$$

where $\mathcal {H}(t)$ is the Heaviside function (4). Then B_i are both mean zero with the covariance given by

$$\begin{equation*} {\rm Cov}(B_{1}, B_{2}) = \frac{2}{\pi }\arcsin (r). \end{equation*}$$

To justify the latter statement, one may identify the explicit expression for Cov(B₁, B₂) as a special integral whose value is known. As an alternative, one may expand Cov(B₁, B₂) into a Taylor series as a function of r around r = 0 via the Hermite expansion for $\mathcal {H}$ as in [11], and note that the coefficients in the expansion are identical to the Taylor coefficients of the arcsine.

Using definition (3) of the defect, we may exchange the order of taking the expectation and integrating to write

$$\begin{eqnarray} &&\fl {\rm Var}(\mathcal {D}_{l}) =\mathbb {E}\big[\mathcal {D}_{l}^{2}\big]=\mathbb {E} \int\!\!\!\int \limits _{\mathcal {S}^{2}\times \mathcal {S} ^{2}}\mathcal {H}(f_{l}(x))\mathcal {H}(f_{l}(y))\,{\rm d}x\,{\rm d}y \nonumber\\ &&\hphantom{\fl {\rm Var}(\mathcal {D}_{l}) }=\int\!\!\!\int \limits _{\mathcal {S}^{2}\times \mathcal {S}^{2}}\mathbb {E}\left[ \mathcal {H}(f_{l}(x))\mathcal {H}(f_{l}(y))\right] {\rm d}x\,{\rm d}y=4\pi \int \nolimits _{\mathcal {S}^{2}}\mathbb { E}[ \mathcal {H}(f_{l}(N))\mathcal {H}(f_{l}(x))]\, {\rm d}x, \end{eqnarray} \tag{ 27 }$$

by the isotropic property of the random field f_l, where N is the northern pole. Note that for every $x\in \mathcal {S}^{2}$, f_l(N) and f_l(x) are jointly Gaussian, centred, with unit variance and covariance equal to

$$\begin{equation*} \mathbb {E}\left[ f_{l}(N)\cdot f_{l}(x)\right] =r_{l}(N,x)=P_{l}(\cos \theta ), \end{equation*}$$

where (θ, ϕ) are the spherical coordinates of x; the latter follows from definition (2) of the covariance function. Therefore, as was explained earlier, for every $x,y\in \mathcal { S}^{2}$,

$$\begin{equation*} \mathbb {E}\left[ \mathcal {H}(f_{l}(x))\mathcal {H}(f_{l}(y))\right] =\frac{2}{\pi }\arcsin (P_{l}(\cos \theta )). \end{equation*}$$

We then evaluate the latter integral in (27) in the spherical coordinates as

$$\begin{equation*} \fl {\rm Var}(\mathcal {D}_{l})=8\pi ^{2}\int \nolimits _{0}^{\pi }\frac{2}{\pi } \arcsin (P_{l}(\cos \theta ))\sin \theta\, {\rm d}\theta =16\pi \int \nolimits _{0}^{\pi }\arcsin (P_{l}(\cos \theta ))\sin \theta\, {\rm d}\theta , \end{equation*}$$

which, taking into account l being even (and thus P_l(t) is also even), is statement (13) of the present lemma.

We start by recalling a basic fact on the asymptotic behaviour of Legendre polynomials (see for instance [15]); as usual, we shall denote by J_ν the Bessel function of the first kind.

Lemma 5.1 (Hilb's asymptotics). For any > 0, C > 0, we have

$$\begin{equation} P_{l}(\cos {\theta }) = \left( \frac{\theta }{\sin (\theta )} \right)^{1/2}J_{0}((l+1/2)\theta ) +\delta (\theta ), \end{equation} \tag{ 28 }$$

where

$$\begin{equation} \delta (\theta ) \lesssim \left\lbrace \begin{array}{@{}l@{\quad }l@{}} \theta ^{1/2}l^{-3/2} & \theta > \dsty\frac{C}{l} \\\ms \theta ^2 & 0 < \theta < \dsty\frac{C}{l}, \end{array}\right. \end{equation} \tag{ 29 }$$

uniformly w.r.t. l ⩾ 1, θ ∈ [0, π − ].

Note that we will use lemma 5.1 only for the range $\theta \in [0,\frac{\pi }{2}]$, so that we may forget about the altogether. We also recall that (see again [15])

$$\begin{equation} |J_{0}(\psi )|=O\left( \frac{1}{\sqrt{\psi }}\right) . \end{equation} \tag{ 30 }$$

Lemma 5.2. Let j ⩾ 5 or j = 3. Then

$$\begin{equation} \int \nolimits _{0}^{\pi /2} P_{l} (\cos \theta )^j\sin {\theta }\, {\rm d}\theta = c_{j} \frac{1}{l^2}(1+o_{j}(1)), \end{equation} \tag{ 31 }$$

where the constants c_j are given by

$$\begin{equation} c_{j} = \int \nolimits _{0}^{\infty } \psi J_{0}(\psi )^j\, {\rm d}\psi , \end{equation} \tag{ 32 }$$

the RHS of (32) being absolutely convergent for j ⩾ 5 and conditionally convergent for j = 3. Moreover, for every j as above, the constants c_j ⩾ 0 are nonnegative, and c₃ > 0 is positive, given explicitly by

$$\begin{equation} c_{3} = \frac{2}{\pi \sqrt{3}}. \end{equation} \tag{ 33 }$$

Proof. By Hilb's asymptotics, we have

$$\begin{equation} \fl \int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )^{j}\sin {\theta }\,{\rm d}\theta =\int \nolimits _{0}^{\pi /2}\left( \left( \frac{\theta }{\sin (\theta )}\right) ^{1/2}J_{0}((l+1/2)\theta )+\delta (\theta )\right) ^{j}\sin {\theta }\, {\rm d}\theta . \end{equation} \tag{ 34 }$$

The contribution of the error term to (34) is (exploiting $\frac{\theta }{\sin \theta }$ being bounded)

$$\begin{equation} \lesssim \int \nolimits _{0}^{\pi /2}|J_{0}((l+1/2)\theta )|^{j-1}\delta (\theta )\theta\, {\rm d}\theta =\int \nolimits _{0}^{1/l}+\int \nolimits _{1/l}^{\pi /2}. \end{equation} \tag{ 35 }$$

Now

$$\begin{equation*} \int \nolimits _{0}^{1/l}|J_{0}((l+1/2)\theta )|^{j-1}\delta (\theta )\theta\, {\rm d}\theta \lesssim \int \nolimits _{0}^{1/l}\theta ^{3}\,{\rm d}\theta \lesssim \frac{1}{l^{4}}, \end{equation*}$$

and using (30), we may bound the second integral in (35) as

$$\begin{equation*} \int \nolimits _{1/l}^{\pi /2}|J_{0}((l+1/2)\theta )|^{j-1}\delta (\theta )\theta\, {\rm d}\theta \lesssim \frac{1}{l^{(j+2)/2}}\int \nolimits _{1/l}^{\pi /2}\theta ^{3/2}{\rm d}\theta \lesssim \frac{1}{l^{(j+2)/2}}. \end{equation*}$$

Plugging the last couple of estimates into (35) , and finally into (34), we obtain for j ⩾ 3

$$\begin{equation} \fl \int \nolimits _{0}^{\pi /2}P_{l}(\cos \theta )^{j}\sin {\theta }\,{\rm d}\theta =\int \nolimits _{0}^{\pi /2}\left( \frac{\theta }{\sin (\theta )}\right) ^{j/2-1}J_{0}((l+1/2)\theta )^{j}\theta\, {\rm d}\theta +O\left( \frac{1}{l^{5/2}} \right) . \end{equation} \tag{ 36 }$$

Therefore, we are to evaluate

$$\begin{equation*} \fl \int \nolimits _{0}^{\pi /2}\left( \frac{\theta }{\sin (\theta )}\right) ^{j/2-1}J_{0}((l+1/2)\theta )^{j}\theta\, {\rm d}\theta =\frac{1}{L^{2}} \int \nolimits _{0}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1}J_{0}(\psi )^{j}\psi\, {\rm d}\psi , \end{equation*}$$

where we denote $L:=l+\frac{1}{2}$ for brevity. The statement of the present lemma is then equivalent to

$$\begin{equation} \int \nolimits _{0}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1}J_{0}(\psi )^{j}\psi\, {\rm d}\psi \rightarrow c_{j}, \end{equation} \tag{ 37 }$$

where c_j is defined by (32).

The main idea is that the main contribution comes from the intermediate range 1 ≲ ψ ⩽ L for any > 0; here we may replace the factor

$$\begin{equation*} \left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1} \end{equation*}$$

with 1. To make this argument precise, we write for $\psi \in \big[0, \frac{\pi }{2}\cdot L\big]$

$$\begin{equation*} \frac{\psi /L}{\sin (\psi /L)}=1+O\left( \frac{\psi ^{2}}{L^{2}}\right) , \end{equation*}$$

so that also

$$\begin{equation} \left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1}=1+O_{j}\left( \frac{ \psi ^{2}}{L^{2}}\right) . \end{equation} \tag{ 38 }$$

Therefore, the integral in (37) is

$$\begin{eqnarray} &&\fl \int \nolimits _{0}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1}J_{0}(\psi )^{j}\psi\, {\rm d}\psi =\int \nolimits _{0}^{L\pi /2}J_{0}(\psi )^{j}\psi\, {\rm d}\psi\nonumber\\ &&+\,O\left( \frac{1}{L^{2}}\int \nolimits _{0}^{L\pi /2}\psi ^{3}|J_{0}(\psi )|^{j}\,{\rm d}\psi \right) . \end{eqnarray} \tag{ 39 }$$

Note that as l → ∞ (equivalently L → ∞), the main term on the RHS of converges to c_j; indeed

$$\begin{equation} \int \nolimits _{0}^{L\pi /2}J_{0}(\psi )^{j}\psi\, {\rm d}\psi \rightarrow c_{j}, \end{equation} \tag{ 40 }$$

so that it remains to bound the error term. Now

$$\begin{equation} \int \nolimits _{0}^{L\pi /2}\psi ^{3}|J_{0}(\psi )^{j}|\,{\rm d}\psi =\int _{0}^{1}+\int _{1}^{L\pi /2}=O(1)+\int _{1}^{L\pi /2}, \end{equation} \tag{ 41 }$$

and we use (30) to bound the latter as

$$\begin{equation*} \int _{1}^{L\pi /2}\psi ^{3}|J_{0}(\psi )^{j}|\,{\rm d}\psi \lesssim \int \nolimits _{1}^{L\pi /2}\psi ^{3-j/2}\,{\rm d}\psi =O(1+l^{4-j/2}), \end{equation*}$$

so that upon plugging the latter into (41) yields

$$\begin{equation*} \int \nolimits _{0}^{L\pi /2}\psi ^{3}|J_{0}(\psi )^{j}|\,{\rm d}\psi =O(1+l^{4-j/2}); \end{equation*}$$

plugging the latter into (39) yields

$$\begin{equation*} \fl \int \nolimits _{0}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{j/2-1}J_{0}(\psi )^{j}\psi\, {\rm d}\psi =\int \nolimits _{0}^{L\pi /2}J_{0}(\psi )^{j}\psi\, {\rm d}\psi +O(l^{-2}+l^{2-j/2}), \end{equation*}$$

so that (40) implies (37) for j ⩾ 5, which was equivalent to the statement of the present lemma in this case.

It then remains to prove the result for j = 3, for which we have to work a little harder due to the conditional convergence of integral (32); to treat this technicality we will have to exploit the oscillatory behaviour of the Bessel function, and not only its decay (30). It is well known that

$$\begin{equation*} J_{0}(\psi )=\sqrt{\frac{2}{\pi }}\frac{\cos (\psi -\pi /4)}{\sqrt{\psi }} +O\left( \frac{1}{\psi ^{3/2}}\right) , \end{equation*}$$

so that

$$\begin{equation} J_{0}(\psi )^3=\left( \frac{2}{\pi }\right) ^{3/2}\frac{\cos (\psi -\pi /4)^{3}}{\psi ^{3/2} }+O\left( \frac{1}{\psi ^{5/2}}\right) , \end{equation} \tag{ 42 }$$

and hence the integral on the RHS of (32) is indeed convergent, by integration by parts.

Now we choose a (large) parameter K ≫ 1 and divide the integration range into [0, K] and $\big[K,L\frac{\pi }{2}\big]$; the main contribution comes from the first term, whence we need to prove that the latter vanishes. Indeed, we use (42) to bound

$$\begin{equation} \int \nolimits _{K}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{1/2}J_{0}(\psi )^{3}\psi\, {\rm d}\psi \lesssim \frac{1}{\sqrt{K}}, \end{equation} \tag{ 43 }$$

where we use integration by parts with the bounded function

$$\begin{equation*} I(T)=\int \nolimits _{0}^{T}\cos (t)^{3}\,{\rm d}t \end{equation*}$$

to bound the contribution leading term in (42); it is easy to bound the contribution of the error term in (42) using the crude estimate (30).

On [0, K], we use (38) to write

$$\begin{equation} \int \nolimits _{0}^{K}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{1/2}J_{0}(\psi )^{3}\psi\, {\rm d}\psi =\int \nolimits _{0}^{K}J_{0}(\psi )^{3}\psi\, {\rm d}\psi +O\left( \frac{K^{5/2}}{l^{2}}\right) , \end{equation} \tag{ 44 }$$

the former clearly being convergent to c₃. Combining (43) with (44), we obtain

$$\begin{equation*} \fl \int \nolimits _{K}^{L\pi /2}\left( \frac{\psi /L}{\sin (\psi /L)}\right) ^{1/2}J_{0}(\psi )^{3}\psi\, {\rm d}\psi =\int \nolimits _{0}^{K}J_{0}(\psi )^{3}\psi\, {\rm d}\psi +O\left( \frac{1}{\sqrt{K}}+\frac{K^{5/2}}{l^{2}}\right) ; \end{equation*}$$

this implies (37) for j = 3 (which is equivalent to the statement of the present lemma for j = 3) upon choosing the parameter K growing to infinity sufficiently slowly (e.g. $K=\sqrt{l}$). Since j = 3 was the only case that was not covered earlier, this concludes the proof of the moment part of lemma 5.2.

It then remains to prove the nonnegativity statement of c_j, and the explicit expression (33) for c₃. In fact, both of those statements follow from the computation we performed in our previous paper [11], p 18. We report the relevant results in lemma 5.3 below.

We recall an alternative characterization for moments of Legendre polynomials from [11]. For given positive integers l₁, l₂, l₃, we introduce the so-called Clebsch–Gordan coefficients $\big\lbrace C_{l_{1}0l_{2}0}^{l_{3}0}\big\rbrace$, which are different from zero if and only if l₁, l₂, l₃ are such that l₁ + l₂ + l₃ is even and l_i + l_j ⩽ l_k for all permutations i, j, k = 1, 2, 3. The Clebsch–Gordan coefficients are well known in group representation theory (they intertwine alternative representations for SO(3)) and in the quantum theory of angular momentum; we do not provide more details here, but refer instead to [11] or to standard references such as [16, 17] (see also [9]). The results provided in [11] are as follows.

Lemma 5.3 ([12], appendix A). For all even l, we have

$$\begin{equation*} \int \nolimits _{0}^{1}P_{l}(t) ^{3}{\rm d}t=\frac{1}{2l+1}\big\lbrace C_{l0l0}^{l0}\big\rbrace ^{2}, \end{equation*}$$

and

$$\begin{equation*} \lim _{l\rightarrow \infty }\frac{1}{2l+1}\left\lbrace C_{l0l0}^{l0}\right\rbrace ^{2}= \frac{2}{\pi \sqrt{3}}. \end{equation*}$$

Also, for j ⩾ 5,

$$\begin{equation*} \int \nolimits _{0}^{1}P_{l}(t)^{j}\,{\rm d}t=\frac{1}{2l+1}\sum _{L_{1}\ldots L_{j-3}}\big\lbrace C_{l0l0}^{L_{1}0}C_{L_{1}0l0}^{L_{2}0}\ldots C_{L_{j-3}0l0}^{l0}\big\rbrace ^{2}>0. \end{equation*}$$

Remark 5.4. The previous discussion yields the following interesting corollary: as l → ∞,

$$\begin{equation*} \lim _{l\rightarrow \infty }\frac{1}{2l+1}\big\lbrace C_{l0l0}^{l0}\big\rbrace ^{2} =\int \nolimits _{0}^{\infty }\psi J_{0}(\psi )^{3}\,{\rm d}\psi , \end{equation*}$$

and for j ⩾ 5,

$$\begin{equation*} \lim _{l\rightarrow \infty }\bigg[ \frac{1}{2l+1}\sum _{L_{1}\ldots L_{j-3}}\big\lbrace C_{l0l0}^{L_{1}0}C_{L_{1}0l0}^{L_{2}0}\ldots C_{L_{j-3}0l0}^{l0}\big\rbrace ^{2} \bigg] =\int \nolimits _{0}^{\infty }\psi J_{0}(\psi )^{j}\,{\rm d}\psi . \end{equation*}$$

The following lemma establishes the lower bound for the constant C in theorem 1.2 claimed in section 1.2. Its proof is straightforward.

Lemma 5.5. For C as in (5), we have

$$\begin{equation*} C>\frac{32}{\sqrt{27}}. \end{equation*}$$

Proof. Recall that $C=\frac{2}{\pi }C_{1}$, where C₁ = C₂ is given by (24). Since all the a_j and c_k are nonnegative, lemma 5.5 follows from bounding the first term in the series

$$\begin{equation*} C\ge 32\pi \times a_{1}c_{3}=\frac{32}{\sqrt{27}}, \end{equation*}$$

where we used a₁ = 1/6 and (33).

Remark 5.6. As a final remark, we note that the moments of P_l(cos θ) are themselves of some physical interest. In particular, while analysing the relationship between asymptotic Gaussianity and ergodicity of isotropic spherical random fields, reference [10] considered the random field

$$\begin{equation*} T(x):=\sum _{l=1}^{\infty }T_{l}(x)=\sum _{l=1}^{\infty }c_{l}Y_{l0}(g\cdot x), \end{equation*}$$

where $Y_{l0}(\theta ,\phi )=\sqrt{\frac{2l+1}{4\pi }}P_{l}(\cos \theta )$ denotes standard spherical harmonics (for m = 0) and g ∈ SO(3) is a uniformly distributed random rotation in $\mathbb {R}^{3}.$ It is readily seen that the resulting field is isotropic, and each of its Fourier components at frequency l is marginally distributed as (for any $x\in \mathcal {S}^{2})$

$$\begin{equation*} Y_{l}=c_{l}\sqrt{\frac{2l+1}{4\pi }}P_{l}(t),\qquad t\sim U[0,1]. \end{equation*}$$

Up to normalization constants, the moments of the marginal law are exactly those of the Legendre polynomials we established earlier in the paper. More precisely, if we focus (as in [10]) on $\widetilde{T} _{l}:=T_{l}(x)/\sqrt{Var(T_{l})},$ we obtain immediately, for any $x\in \mathcal {S}^{2}$

$$\begin{equation*} \mathbb {E}\lbrace \widetilde{T}_{l}(x)\rbrace =\mathbb {E}\lbrace \sqrt{ (2l+1)}P_{l}(t)\rbrace =0,\mathbb {E}\lbrace \widetilde{T} _{l}(x)\rbrace ^{2}=1, \end{equation*}$$

and

$$\begin{equation*} \mathbb {E}\lbrace \widetilde{T}_{l}(x)\rbrace ^{3}=E\lbrace \sqrt{(2l+1)} P_{l}(t)\rbrace ^{3}=O\left(\frac{1}{\sqrt{l}}\right), \end{equation*}$$

whereas [18]

$$\begin{equation*} E\lbrace \sqrt{(2l+1)}P_{l}(t)\rbrace ^{4}\simeq \log l, \end{equation*}$$

and hence all moments of order q ⩾ 4 diverge.

Note that in lemma 5.2, we worked relatively hard to establish the precise asymptotics for the moments of Legendre polynomials. A much cruder version of the same argument gives a uniform upper bound for the fifth moment of the absolute value of the Legendre polynomials.

Lemma 5.7. We have the following uniform upper bound for the fifth moment of the absolute value of Legendre polynomials

$$\begin{equation} \int \nolimits _{0}^{\pi /2}|P_{l}(\cos {\theta })|^{5}\sin \theta {\rm d}\theta =O\left( \frac{1}{l^{2}}\right) , \end{equation} \tag{ 45 }$$

where the constant involved in the `O'-notation is universal.

Proof. The proof follows along the same lines as the beginning of the proof of lemma 5.2 for k ⩾ 5, except that we use the crude upper bound (30) whenever we reach (36), and the boundedness of $\frac{\theta }{\sin {\theta }}$ on $\big[ 0,\frac{ \pi }{2}\big]$.

We wish to thank Zeév Rudnick for suggesting the problem, many stimulating and fruitful discussions and useful comments on an earlier version of this paper, and Mikhail Sodin for many stimulating and fruitful discussions. We are grateful to the anonymous referees for many useful comments on how to improve the presentation of the present work. A substantial part of this research was done during the second author's visit to the University of Rome 'Tor Vergata', and he would like to acknowledge the extremely friendly and stimulating environment in the institution and the generous financial support. He was supported by the Knut and Alice Wallenberg Foundation, grant KAW.2005.0098.