aa r X i v : . [ m a t h . P R ] J u l The Defect ofRandom Hyperspherical Harmonics
Maurizia Rossi
MAP5-UMR CNRS 8145, Universit´e Paris Descartes, FranceE-mail address: [email protected]
Abstract
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d -sphere ( d ≥ Keywords and Phrases: Defect, Gaussian Eigenfunctions, High-EnergyAsymptotics, Quantitative Central Limit Theorem, Integrals of Hyper-spherical HarmonicsAMS Classification: 60G60; 42C10, 60D05, 60B10, 43A75
Let f : M → R be any real-valued function defined on some compact Riemannian manifold( M , g ) and let µ g denote the induced measure on M . The defect D ( f ) of f is the differencebetween the measure of “hot” and “cold” regions: D ( f ) := µ g ( f − (0 , + ∞ )) − µ g ( f − ( −∞ , . We can hence write D ( f ) = Z M H ( f ( x )) dµ g ( x ) , (1.1)where H denotes the Heaviside function H ( t ) := 1 (0 , + ∞ ) ( t ) − ( −∞ , ( t ), t ∈ R .An important case is where f is a Laplacian eigenfunction. We recall that a function f is called a Laplacian eigenfunction if it is a non-trivial solution of the Schr¨odinger equation∆ g f + Ef = 0 , (1.2)where ∆ g stands for the Laplace-Beltrami operator on ( M , g ) and E >
0. It is well-knownthat the (purely discrete) spectrum of − ∆ g consists of a non-decreasing sequence of positiveeigenvalues whose corresponding sequence of eigenfunctions forms a complete orthonormalbasis for L ( M ), the space of square integrale functions on the manifold. Observe that weallow multiple eigenvalues i.e., spectral degeneracies.1n increasing amount of mathematics research has focused on the geometry of the nodalset f − (0) (see e.g. [Br¨u78, BG72, DF88, Yau82]) and its complement M \ f − (0) (seee.g. [GRS13, JZ16]), associated with Laplacian eigenfunctions f . Note that nodal sets arecustomarly called “nodal lines” in the two-dimensional case – being, generically, smoothcurves – and also that the connected components of M \ f − (0) are often referred to as“nodal domains”. The defect (1.1) is one of the most natural functionals [MW11] associatedwith the geometry of the latter.Recently, a growing interest has been attracted by random eigenfunctions on manifolds(see also [Mec09]) - especially on the two-dimensional sphere and the standard flat torus(e.g. [BMW17, KKW13, MPRW16, MRW17, MW11, MW14, NS09, RW08, Wig10]). Inthe latter references, the space of eigenfunctions is endowed with some probability measureand the geometry of their (random) nodal sets and domains is studied (in the high-energylimit, i.e. when the magnitude of the eigenvalues diverges to infinity). See § Some conventions.
In this manuscript, given two sequences a n , b n of positive numbers,we will write a n ∼ b n if lim n → + ∞ a n /b n = 1, whereas a n = O ( b n ) or equivalently a n ≪ b n (resp. a n = o ( b n )) if a n /b n is asymptotically bounded (resp. a n /b n → a n ≍ b n will mean that a n /b n → c , for some c >
0. Every random object will be defined on thesame probability space (Ω , F , P ), E shall denote the expectation under the measure P and,as usual, → L convergence in distribution whereas = L equality in law. We deal with the case M = S d ֒ → R d +1 , the unit d -dimensional sphere with the naturalmetric ( d ≥ dx . The eigenvalues of theLaplace-Beltrami operator on S d (which will be denoted by ∆ d from now on), are of theform − ℓ ( ℓ + d − ℓ ∈ N , and the dimension n ℓ ; d of the ℓ -th eigenspace is n ℓ ; d = 2 ℓ + d − ℓ (cid:18) ℓ + d − ℓ − (cid:19) ∼ d − ℓ d − , as ℓ → + ∞ . An orthonormal basis for the ℓ -th eigenspace is given by the family of (real-valued) hyper-spherical harmonics ( Y ℓ,m ; d ) n ℓ ; d m =1 (see e.g. [VK93, § d Y ℓ,m ; d + ℓ ( ℓ + d − Y ℓ,m ; d = 0 . We now endow the ℓ -th eigenspace with a Gaussian measure, i.e. we consider the ℓ -th(real-valued) random eigenfunction T ℓ := T ℓ ; d on S d to be defined as T ℓ ( x ) := n ℓ ; d X m =1 a ℓ,m ; d Y ℓ,m ; d ( x ) , x ∈ S d , (1.3)where ( a ℓ,m ; d ) n ℓ ; d m =1 are i.i.d. centered Gaussian random variables with variance given byVar( a ℓ,m ; d ) = | S d | n ℓ ; d , (1.4) | S d | denoting the (Lebeasgue) measure of the hyperspherical surface. Equivalently, we candefine T ℓ as the isotropic centered Gaussian random field on S d whose covariance kernel isCov ( T ℓ ( x ) , T ℓ ( y )) = G ℓ ; d (cos d ( x, y )) , x, y ∈ S d , (1.5)2here G ℓ ; d stands for the normalized ℓ -th Gegenbauer polynomial [Sze75, § d ( x, y )denotes the (spherical) geodesic distance between x and y .To be more precise, G ℓ ; d = α − ℓ P ( d/ − ,d/ − ℓ , where (cid:16) P ( a,b ) ℓ (cid:17) ℓ denotes the family ofJacobi polynomials [Sze75, Chapter 4] and α ℓ = (cid:0) ℓ + d/ − ℓ (cid:1) is a normalizing factor. It turnshence out that T ℓ ( x ) has unit variance for every x ∈ S d .This model was studied in [MR15] and, in the particular case d = 2 in [BMW17, CM16,CMW16, MP11, MRW17, MW11, MW14, NS09, Wig10] e.g. Note that when d = 2, (1.4)is Var( a ℓ,m ;2 ) = π ℓ +1 and G ℓ ;2 ≡ P ℓ the ℓ -th Legendre polynomial [Sze75, § § | S d | n ℓ ; d n ℓ ; d X m =1 Y ℓ,m ; d ( x ) Y ℓ,m ; d ( y ) = G ℓ ; d (cos d ( x, y )) , x, y ∈ S d (1.6)ensures that the random field T ℓ as defined in (1.3) has covariance kernel given by (1.5).The defect D ℓ := D ( T ℓ ) in (1.1) of T ℓ is then a random variable defined as D ℓ = Z S d H ( T ℓ ( x )) dx. (1.7)We are interested in the asymptotic behavior of the sequence ( D ℓ ) ℓ in the high-energy limit,i.e. as ℓ → + ∞ . We anticipate here that D ℓ vanishes for odd ℓ , therefore we will study thedefect only for even integers ℓ (we will prove it in § ℓ → + ∞ weshall mean: as ℓ → + ∞ along even integers . The case d = 2 has been investigated by Marinucci and Wigman. In [MW11], they provethat D ℓ is centered and give an asymptotic result for the variance, i.e. as ℓ → + ∞ Var( D ℓ ) = Cℓ (1 + o (1)) , C > √ . (1.8)In [MW14], a Central Limit Theorem is shown for the defect on the 2-sphere: as ℓ → + ∞ D ℓ p Var( D ℓ ) L → Z, (1.9)where Z ∼ N (0 ,
1) is a standard Gaussian random variable.Observe now that a simple transformation gives D ℓ = 2 Z S d (0 , + ∞ ) ( T ℓ ( x )) dx − | S d | , where R S d (0 , + ∞ ) ( T ℓ ( x )) dx =: S ℓ (0) is the measure of the so-called 0-excursion set { x ∈ S d : T ℓ ( x ) > } . The general case of z -excursion set for z ∈ R , on the d -sphere ( d ≥
2) has beenstudied in [MR15]. In the latter reference, quantitative CLTs in the Wasserstein distancefor the measure S ℓ ( z ) := R S d ( z, + ∞ ) ( T ℓ ( x )) dx of z -excursion sets { x ∈ S d : T ℓ ( x ) > z } aregiven (see below (1.11) and (1.12)), except for the nodal case z = 0. Recall that Wasserstein Recall that (cid:16) P ( a,b ) ℓ (cid:17) ℓ is a family of orthogonal polynomials on the interval [ − ,
1] with respect to theweight (1 − t ) a (1 + t ) b . § C.2]) is the probability metric between two random variables
N, Z defined as d W ( N, Z ) := sup h ∈ Lip | E [ h ( N )] − E [ h ( Z )] | , (1.10)where Lip denotes the set of Lipschitz functions whose Lipschtiz constant equals 1.From [MR15] for z = 0, we have thatVar( S ℓ ( z )) ∼ | S d | ( d − z φ ( z ) × ℓ d − , as ℓ → + ∞ , (1.11) φ (resp. Φ) denoting the standard Gaussian density (resp. distribution function), andmoreover d W S ℓ ( z ) − | S d | (1 − Φ( z )) p Var( S ℓ ( z )) , Z ! = O (cid:18) √ ℓ (cid:19) , (1.12)where Z ∼ N (0 ,
1) as before. In particular, (1.12) implies a CLT for the measure ofexcursion sets at any non-zero level.
In this paper we study the high-energy behavior of the sequence of random variables ( D ℓ ) ℓ (1.7) in any dimension d ≥
2. Evaluating the mean of D ℓ is trivial. Indeed, exchanging theexpectation with integration over S d we get E [ D ℓ ] = Z S d E [ H ( T ℓ ( x ))] dx, and since for every x ∈ S d , E [ H ( T ℓ ( x ))] = 0 by the symmetry of the Gaussian distribution,we have just proved the following. Lemma 1.1.
For every ℓ ∈ N E [ D ℓ ] = 0 . For the variance, we have the following asymptotic result which generalizes (1.8) to thehigher dimensional sphere and whose proof is given in § Proposition 1.2. As ℓ → + ∞ , the defect variance Var( D ℓ ) satisfies Var( D ℓ ) = C d ℓ d (1 + o (1)) , (1.13) where C d > is a positive constant depending only on d . Note that C = C in (1.8). Comparing (1.13) with (1.11), one infers that the variance ofthe measure of z -excursion sets has a smaller order of magnitude in the nodal case than for z = 0. This phenomenon appears in many situations and it is usually referred to as Berry’scancellation phenomenon [KKW13, Wig10]. See § C d in Proposition 1.2 may be expressed in terms of the improper (condi-tionally convergent) integral C d = 4 π | S d || S d − | Z + ∞ ψ d − (cid:16) arcsin (cid:16) e J d ( ψ ) (cid:17) − e J d ( ψ ) (cid:17) dψ, (1.14)4here e J d ( ψ ) := 2 d/ − ( d/ − J d/ − ( ψ ) ψ − ( d/ − , ψ > J d/ − denotes the Bessel function [Sze75, § d/ −
1. See (4.38) for a formulaequivalent to (1.14) that expresses C d as a convergent series. We do not know whether onecan evaluate C d explicitly; however, we shall show that C d > π | S d || S d − | (cid:16) d/ − ( d/ − (cid:17) d/ − / d/ − − √ π Γ ( d/ − / , (1.16)Γ denoting the Gamma function [Sze75, § d = 2, (1.16) gives C > / √
27, that coincides with (1.8).The main contribution of the present paper is the following quantitative CLT in theWasserstein distance (1.10) which extends and generalizes the results from [MR15, MW14]collected in § § § not enough to dealwith the defect, and we have to overcome some additional difficulties. Finally, our resultis stronger than (1.9) (proven in [MW14]), yielding also the rate of convergence to theGaussian distribution. Theorem 1.3.
Let Z be a standard Gaussian random variable. For d ≥ we have, as ℓ → + ∞ , d W D ℓ p Var( D ℓ ) , Z ! = O (cid:18) √ log ℓ (cid:19) , in particular D ℓ p Var( D ℓ ) L −→ Z. The proof of Theorem 1.3 will be given in § d = 2 and d > d = 3 , , § H the third Hermite polynomial, i.e. H ( t ) = t − t, t ∈ R , and by d D either the Wasserstein (1.10), Kolmogorov or Total Variation distance (see [NP12, § C.2]),then
Lemma 1.4.
For d ≥ , as ℓ → + ∞ d D R S d H ( T ℓ ( x )) dx q Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) , Z = O vuut cum (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) = O (cid:18) √ ℓ d − (cid:19) , (1.17) where cum (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) denotes the fourth cumulant of R S d H ( T ℓ ( x )) dx . (Note that the first equality in (1.17) is an application of the so-called Fourth MomentTheorem [NP12, Theorem 5.2.7].) In particular, Lemma 1.4 entails that, as ℓ → + ∞ , R S d H ( T ℓ ( x )) dx q Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) L −→ Z, see [PT11, (3.1.3)] Z is a standard Gaussian random variable. Note that, for d ≥
3, (2.21) in [MR15,Proposition 2.3] gives only O (cid:0) ℓ − ( d − / (cid:1) for the l.h.s. of (1.17) - which does not vanishwhen d ∈ { , , } , as ℓ → + ∞ .The random variable R S d H ( T ℓ ( x )) dx is the so-called bispectrum of T ℓ and it is of in-dependent interest (see [Mar06, Mar08]). In particular for d = 2, the information on thebispectrum of T ℓ are used to test some features of Cosmic Microwave Background [MP11];on the 2-sphere, (1.17) coincides with the result found by Marinucci in [Mar08].Our argument in order to prove Lemma 1.4 requires technical computations involvingconcatenated sums of integrals of three hyperspherical harmonics of the form G ℓ,m ℓ,m ,ℓ,m ; d := Z S d Y ℓ,m ; d ( x ) Y ℓ,m ; d ( x ) Y ℓ,m ; d ( x ) dx, (1.18)for m , m , m ∈ { , , . . . , n ℓ ; d } ; (note that by definition, G ℓ,m ℓ,m ,ℓ,m ; d is invariant underany permutations of indexes m , m , m ). The integral G ℓ,m ℓ,m ,ℓ,m ; d is strictly related to so-called Clebsch-Gordan coefficients [Far08, MP11, VK93] for the special orthogonal group SO ( d + 1), which play a key role in group representation properties of the latter.The integral in (1.18) is well-known for d = 2 (so-called Gaunt formula - see [MP11,Proposition 3.43]) and several applications by many authors can be found (see [CM15,Mar06, Mar08, MP11, MW14] e.g.), because of its importance also in the quantum theoryof angular momentum. From [VMK88, (5.6.2.12),(5.6.2.13)], for the two-dimensional case, G ℓ,m ℓ,m ,ℓ,m ;2 = r ℓ + 14 π C ℓ,m ℓ,m ,ℓ,m · C ℓ, ℓ, ,ℓ, , (1.19)where C ℓ ,m ℓ ,m ,ℓ ,m denote Clebsch-Gordan coefficients (see [VMK88, Chapter 8] or [MP11, § SO (3); explicit formulas are known for the latter. (Note that oneusually considers m ∈ {− ℓ, . . . , ℓ } instead of m ∈ { , , . . . , ℓ + 1 } , see e.g. [MP11, § novel tools in order to complete our argument.The main achievement in this direction is the following result whose proof, given in § S d , a topic which has recently received some attention (see e.g. [Dur16]). Lemma 1.5.
For every even ℓ ∈ N , M, M ′ ∈ { , , . . . , n ℓ ; d } and d ≥ n ℓ ; d X m ,m =1 G ℓ,Mℓ,m ,ℓ,m ; d G ℓ,M ′ ℓ,m ,ℓ,m ; d = δ M ′ M ( n ℓ ; d ) | S d | | S d − || S d | Z − G ℓ ; d ( t ) (cid:16)p − t (cid:17) d − dt. In the case d = 2, from (1.19), by orthonormality properties of Clebsch-Gordan coefficients[MP11, (3.62)] and since (2 ℓ + 1) − (cid:16) C ℓ, ℓ, ,ℓ, (cid:17) = R − P ℓ ( t ) dt (again from (1.19), see also[MW14]), we have X m ,m G ℓ,Mℓ,m ,ℓ,m ;2 G ℓ,M ′ ℓ,m ,ℓ,m ;2 = δ M ′ M (2 ℓ + 1) · π Z − P ℓ ( t ) dt, which coincides with the statement of Lemma 1.5.6 .4 Plan of the paper In § § § § § §
5, while § § This topic was suggested by Domenico Marinucci. The author would like to thank him,Giovanni Peccati and Igor Wigman for useful conversations, and an anonymous referee forinsightful comments.The research leading to this work was carried out within the framework of the ERC
Pascal project no. 277742 and of the Grant STARS (R-AGR-0502-10) at Luxembourg University.The author is currently supported by the Fondation Sciences Math´ematiques de Paris andthe ANR-17-CE40-0008 project
Unirandom . Berry argued that, at least for “generic” chaotic surfaces, the local behavior of (determinis-tic) eigenfunctions should be universal [Ber77]. He proposed to compare the eigenfunction f in (1.2) of large eigenvalue E to a “typical” instance of a monochromatic random wave withwavenumber √ E ; we can define the latter as the centered Gaussian field W = ( W ( x )) x ∈ R on the plane, whose covariance structure is given byCov ( W ( x ) , W ( y )) = J (cid:16) √ E | x − y | (cid:17) , x, y ∈ R , (2.20) J being the 0-order Bessel function [Sze75, § f can then be pre-dicted by W ; for instance, nodal lines of the latter should model nodal lines of f (see[Wig12]).In the spherical case, from (1.5) with d = 2, the random model (1.3) has covariancekernel given by Cov ( T ℓ ( x ) , T ℓ ( y )) = P ℓ (cos d ( x, y )), x, y ∈ S , where P ℓ ≡ G ℓ ;2 is still the ℓ -th Legendre polynomial [Sze75, § P ℓ (cos θ ) ∼ r θ sin θ J (( ℓ + 1 / θ )uniformly for θ ∈ [0 , π − ε ], similar to (2.20) but for the square root that keeps a trace aboutthe geometry of the sphere. In recent years, the geometry of random spherical eigenfunctionshas been studied by several papers, motivated also by applications in Cosmology - forinstance concerning the analysis of CMB [MP11]. In particular, so-called Lipschitz Killingcurvatures [AT07, § d = 2), the boundarylength [Ros15, Wig10, Wig12, MRW17], the area [MR15, MW11, MW14] and the Euler-Poincar´e characteristic [CM16, CMW16] of excursion sets at any level. For each of the justmentioned geometric functionals, the same qualitative behavior has been observed, i.e. alower-order asymptotic variance in the nodal case (see [CM16, § z = 0. This phenomenon has a deeper interpretation related to chaoticexpansions which also explain the rate we obtain in the nodal case (Theorem 1.3 comparedto (1.12)); we shall be back on this issue in § nice compactmanifold (as the multidimensional torus T d := R d / Z d ). Our approach to establish Proposition 1.2 and Theorem 1.3 relies, among other things, onthe chaotic decomposition of D ℓ (1.7) (this technique was used also in [MW14, MPRW16,MRW17] e.g.). Since the defect is a square integrable fuctional of a Gaussian field, it canbe written as a series D ℓ = P + ∞ q =0 D ℓ [ q ] , converging in L ( P ), where D ℓ [ q ] is the orthogonalprojection of D ℓ onto the so-called q -th Wiener chaos ( D ℓ [0] = E [ D ℓ ]); the random variables D ℓ [ q ], q ≥ § § H k ) k ≥ are defined as H ( t ) = 1 and for k ≥ H k ( t ) := ( − k φ ( t ) − d k dt k φ ( t ) , t ∈ R , where φ still denotes the probability density of a standard Gaussian random variable. Forinstance, H ( t ) = t , H ( t ) = t − H ( t ) = t − t and so on. Lemma 2.1.
The Wiener-Itˆo chaos decomposition of the defect D ℓ is D ℓ = + ∞ X q =1 D ℓ [2 q + 1] = + ∞ X q =1 J q +1 (2 q + 1)! Z S d H q +1 ( T ℓ ( x )) dx, (2.21) where J q +1 := 2 √ π H q (0) . The first marked difference compared to the non-zero level case treated in [MR15] is thatthe non-linear functionals investigated in the latter reference are assumed to have non -vanishing (and asymptotically leading) second chaotic component, whereas from (2.21) wehave D ℓ [2] = 0 for every ℓ . Some of the tools used in [MR15] are hence not suitable, andwe will need some new technical results. We will clarify this point in what follows. Our argument to prove Proposition 1.2 is essentially equivalent to the one obtained bygeneralizing to higher dimensional setting the approach used in [MW11] to find the defectasymptotic variance on the 2-sphere.Let us keep in mind (2.21). By virtue of the orthogonality and isometric properties ofchaotic projections and some straightforward computations, we have thatVar( D ℓ ) = | S d || S d − | + ∞ X q =1 J q +1 (2 q + 1)! Z π G ℓ ; d (cos ϑ ) q +1 (sin ϑ ) d − dϑ ; (2.22)8ince Gegenbauer polynomials are symmetric, that is, G ℓ ; d ( − t ) = ( − ℓ G ℓ ; d ( t ), t ∈ [ − , § ℓ . For even ℓ , we haveVar( D ℓ ) = 2 + ∞ X q =1 J q +1 (2 q + 1)! | S d || S d − | Z π/ G ℓ ; d (cos ϑ ) q +1 (sin ϑ ) d − dϑ. (2.23)Lemma 1.1 hence implies that D ℓ = 0 for odd ℓ , as anticipated in § ℓ → + ∞ , Z π/ G ℓ ; d (cos ϑ ) q +1 (sin ϑ ) d − dϑ = c q +1; d ℓ d (1 + o (1)) , for some nonnegative constant c q +1; d (in [MR15, MW11] it was conjectured that c q +1; d > q and d ). One would hence expect Proposition 1.2 to hold with C d = 2 | S d || S d − | + ∞ X q =1 J q +1 (2 q + 1)! c q +1; d . In § Z π G ℓ ; d (cos ϑ ) (sin ϑ ) d − dϑ = | S d || S d − | n ℓ ; d , in particular it is of bigger order, as ℓ → + ∞ , than the q -th moment for any q ≥
3. Thismake easier the investigation of the variance of non-linear functionals whose second chaoticcomponent is non-zero and asymptotically leading (as those considered in [MR15]). Indeed,in the latter case the leading term for the asymptotic variance corresponds to the secondchaotic projection, whereas for the defect all terms equally contribute.
As a consequence of what was mentioned just above in § one chaotic term (the second). Since the latter is proportional to a sum of independentrandom variables, an application of the standard Central Limit Theorem allows to concludethe investigation. For the defect instead, we have to study the asymptotic behavior of everychaotic component.The CLT recalled in (1.9) and (1.12) make it plausible to conjecture that the defect isasymptotically Gaussian in any dimension; we wish hence to extend and generalize (1.9)and (1.12). To prove a CLT for the series (2.21), we first need to deal with single chaoticcomponents. Theorem 1.2 in [MR15] ensures that D ℓ [2 q + 1] is, as ℓ → + ∞ , Gaussian forevery pair (2 q + 1 , d ) except for (3 , d ) when d = 3 , ,
5. It is however reasonable to believethat also D ℓ [3] is asymptotically normal in any dimension, as Lemma 1.4 states.To prove the latter, since we are in a fixed chaos - the third one - we can use FourthMoment Theorem [NP12, Theorem 5.2.7] i.e., to have asymptotic Gaussianity it is enough(and necessary) that cum (cid:0)R S d H ( T ℓ ( x )) dx (cid:1)(cid:0) Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1)(cid:1) → , (2.24)9here by cum ( X ) we denote the 4-th cumulant [PT11, (3.1.3)] of the random variable X .Note that from Thorem 5.2.6 in [NP12], the l.h.s. of (2.24) allows to estimate moreoverthe rate of convergence to the Gaussian distribution in various probability metrics [NP12, § C.2], Wasserstein distance (1.10) included (see § § (cid:18)Z S d H ( T ℓ ( x )) dx (cid:19) = Z ( S d ) cum (cid:0) H ( T ℓ ( w )) , H ( T ℓ ( z )) , H ( T ℓ ( w ′ )) , H ( T ℓ ( z ′ )) (cid:1) dwdzdw ′ dz ′ . The diagram formula for Hermite polynomials [MP11, Proposition 4.15] applied to theintegrand of the r.h.s. of the last equality and results by Nourdin and Peccati, in partic-ular [NP12, Lemma 5.2.4], ensure that the major contribution for the fourth cumulant of R S d H ( T ℓ ( x )) dx comes from so-called circulant diagrams . Indeed, in [MR15, Lemma 4.1]it has been shown that the contribution of the latter can be expressed as multiple integralsof products of powers of Gegenbauer polynomials. We are then left in our case with thefollowing I ℓ ; d := Z ( S d ) G ℓ ; d (cos d ( w, z )) G ℓ ; d (cos d ( w, w ′ )) × G ℓ ; d (cos d ( w ′ , z ′ )) G ℓ ; d (cos d ( z ′ , z )) dwdzdw ′ dz ′ . (2.25)In [MR15, Proposition 4.2, Proposition 4.3] to prove (2.24), upper bounds for (2.25) aregiven which are too big for d = 3 , ,
5: the technique consists first of bounding the contri-bution of one of the involved Gegenbauer polynomials by its sup-norm (= 1) and then ofusing Cauchy-Schwarz inequalities. This reduces to deal simply with moments of Gegen-bauer polynomials but allows to get only not satisfactory bounds. Our argument is subtler and more difficult, since we have to compute the exact asymptotics for I ℓ ; d in (2.25), as ℓ → + ∞ .The addition formula (1.6) applied several times in (2.25), leads to concatenated sums ofintegrals of three hyperspherical harmonics. With the same notation as in (1.18) we find I ℓ ; d = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ,m ,m ′ ,m ′ ,m ′ =1 G ℓ,m ℓ,m ′ ,ℓ,m ′ ; d G ℓ,m ′ ℓ,m ,ℓ,m ; d G ℓ,m ℓ,m ,ℓ,m ; d G ℓ,m ′ ℓ,m ′ ,ℓ,m ′ ; d . The idea now is to find some useful property for double sums of these coefficients (1.18) (asdone in Lemma 1.5). Once Lemma 1.4 is proved, then an argument similar to the one givenin the proof of Corollary 4.2 in [MW14], provides a CLT for the defect in any dimension. Weare however interested in subtler results: rates of convergence to the limiting distribution -as explained in the next section.
In order to obtain rates of convergence, i.e. to prove our Theorem 1.3, it is not enough togeneralize to higher dimensional setting the approach used in [MW14]. Indeed in particularone needs refined estimates on the distance in distribution between the “truncated” defect(see just below) and the Gaussian law. 10et us truncate the series (2.21) at some frequency m , obtaining D mℓ = P mq =1 D ℓ [2 q + 1].We know that D mℓ is Gaussian, as ℓ → + ∞ , (since it is a linear combination of asymp-totically normal random variables living in different order chaoses; see [PT05]) and we canthen compute the rate of convergence in Wasserstein distance (1.10) to the limiting distri-bution by using Stein-Malliavin techniques for Normal approximations [NP12, Chapters 5,6]. The tail D ℓ − D mℓ can be controlled by its L ( P )-norm. Summing up all contributionsand choosing an optimal speed m = m ( ℓ ), we obtain Theorem 1.3.The rate of convergence is slower than in the non-nodal case (1.12). Indeed, here allchaoses in the Wiener-Itˆo expansion for the defect (2.21) contribute, whereas for z = 0 thesecond chaos does not vanish and hence gives the dominating term, making easier also theestimation of the speed of convergence. For the defect instead, one has to control the wholeseries.As remarked also in [CM16, § § For a complete discussion on the following topics see [NP12].
Let X = ( X ( f )) f ∈ L ( S d ) be an isonormal Gaussian process on L ( S d ), the space of squareintegrable functions on the d -sphere ( d ≥ L ( S d ) verifying the isometric property, i.e.Cov ( X ( f ) , X ( h )) = h f, h i L ( S d ) := Z S d f ( x ) h ( x ) dx, f, h ∈ L ( S d ) . We can construct it as follows. Let us denote by B ( S d ) the Borel σ -field on S d and let W = { W ( A ) : A ∈ B ( S d ) } be a centered Gaussian family on S d such thatCov ( W ( A ) , W ( B )) = Z S d A ∩ B ( x ) dx, where 1 A ∩ B denotes the indicator function of the set A ∩ B . The random field X on L ( S d )defined as X ( f ) := Z S d f ( x ) dW ( x ) , f ∈ L ( S d ) , (3.26)i.e. the Wiener-Itˆo integral of f with respect to the Gaussian measure W , is the isonormalGaussian process on the d -sphere.Note that for random eigenfunctions (1.3) it holds T ℓ ( x ) L = X ( f ℓ,x ) , x ∈ S d , (3.27)as stochastic processes, where X is defined as in (3.26) and f ℓ,x := f ℓ,x ; d is given by f ℓ,x ( y ) := r n ℓ ; d | S d | G ℓ ; d (cos d ( x, y )) , y ∈ S d . (3.28)11 .2 Defect in Wiener chaoses Let us now recall the notion of Wiener chaos, mentioned in § H k ) k ≥ denote the sequence of Hermite polynomials [Sze75, § (cid:16) H k √ k ! (cid:17) k ≥ form an orthonormal basis of L ( R , φ ( t ) dt ), the space of square integrable func-tions on the real line endowed with the Gaussian measure. Recall that for jointly Gaussianrandom variables Z , Z ∼ N (0 ,
1) and k , k ≥
0, we have E [ H k ( Z ) H k ( Z )] = k ! ( E [ Z Z ]) k δ k k . (3.29)For each integer q ≥
0, consider the closure C q in L ( P ) of the linear space generated byrandom variables of the form H q ( X ( f )) , f ∈ L ( S d ) , k f k L ( S d ) = 1 . The space C q is the so-called q -th Wiener chaos associated with X . By (3.29), it is easy tocheck that C q ⊥ C q ′ for q = q ′ and moreover the following Wiener-Itˆo chaos decompositionholds L ( P ) = + ∞ M q =0 C q , i.e. every random variable F whose second moment is finite can be expressed as a seriesconverging in L ( P ) F = + ∞ X q =0 F [ q ] , (3.30)where F [ q ] = proj( F | C q ) is the orthogonal projection of F onto the q -th chaos ( F [0] = E [ F ]).The defect D ℓ , as defined in (1.7), is a square integrable functional of the isonormalGaussian process X in (3.26), actually it is easy to check that | D ℓ | ≤ | S d | a.s. It henceadmits a Wiener-Itˆo chaos decomposition of the form (3.30) (see Lemma 2.1). The q -th tensor power L ( S d ) ⊗ q (resp. q -th symmetric power L ( S d ) ⊙ q ) of L ( S d ) is simply L (( S d ) q , dx q ) (resp. L s (( S d ) q , dx q ), i.e. the space of a.e. symmetric functions on ( S d ) q ).Let us define the (linear) operator I q for unit norm f ∈ L ( S d ) as I q ( f ⊗ q ) := H q ( X ( f ))and extend it to an isometry between L s (( S d ) q ) := L s (( S d ) q , dx q ) equipped with the modi-fied norm √ q ! k · k L s (( S d ) q ) and the q -th Wiener chaos C q endowed with the L ( P )-norm.It is well-known [NP12, § h ∈ L s (( S d ) q ), it holds I q ( h ) = Z ( S d ) q h ( x , x , . . . , x q ) dW ( x ) dW ( x ) . . . dW ( x q ) , the multiple Wiener-Itˆo integral of h with respect to the Gaussian measure W , where thedomains of integration implicitly avoids diagonals. Therefore, F [ q ] in (3.30) is a stochasticmultiple integral F [ q ] = I q ( f q ), for a unique kernel f q ∈ L s (( S d ) q ).From Lemma 2.1 and (3.27), (3.28), it is immediate to check that for chaotic projectionsof the defect we have ( q ≥ D ℓ [2 q + 1] = I q +1 (cid:18) J q +1 (2 q + 1)! Z S d f ℓ ; x dx (cid:19) . .2.2 Contractions For every p, q ≥ f ∈ L ( S d ) ⊗ p , g ∈ L ( S d ) ⊗ q and r = 1 , , . . . , p ∧ q , the so-called contraction of f and g of order r is the element f ⊗ r g ∈ L ( S d ) ⊗ p + q − r defined as (see[NP12, (B.4.7)])( f ⊗ r g )( x , . . . , x p + q − r ):= Z ( S d ) r f ( x , . . . , x p − r , y , . . . , y r ) g ( x p − r +1 , . . . , x p + q − r , y , . . . , y r ) dy . . . dy r . (3.31)For p = q = r , we have f ⊗ r g = h f, g i L ( S d ) ⊗ r and for r = 0, f ⊗ g := f ⊗ g . Denote by f e ⊗ r g the canonical symmetrization [NP12, (B.2.1)] of f ⊗ r g . The following multiplication formulais well-known [NP12, Theorem 2.7.10]: for p, q = 1 , , . . . , f ∈ L ( S d ) ⊙ p , g ∈ L ( S d ) ⊙ q , wehave I p ( f ) I q ( g ) = p ∧ q X r =0 r ! (cid:18) pr (cid:19)(cid:18) qr (cid:19) I p + q − r ( f e ⊗ r g ) . In what follows, we will use standard notions and results from Malliavin calculus; we referthe reader to [NP12, § § q, r ≥
1, recall that the r -th Malliavin derivative of a random variable I q ( f ) ∈ C q where f ∈ L ( S d ) ⊙ q , can be identified as the element D r I q ( f ) : Ω → L ( S d ) ⊙ r given by D r I q ( f ) = q !( q − r )! I q − r ( f ) , (3.32)for r ≤ q , and D r I q ( f ) = 0 for r > q . For notational simplicity, we shall write D in-stead of D . We say that F as in (3.30) belongs to the space D r,q if E [ | F | q ] + · · · + E h k D r F k qL ( S d ) ⊙ r i < + ∞ . We write k F k D r,q := (cid:16) E [ | F | q ] + . . . E h k D r F k qL ( S d ) ⊙ r i(cid:17) q . It is easy to check that F ∈ D , if and only if ∞ X q =1 q k F [ q ] k L ( P ) < + ∞ , and in this case E h k DF k L ( S d ) i = P ∞ q =1 q k F [ q ] k L ( P ) . In particular, if F ∈ L ( P ) admitsa finite chaotic decomposition, then it belongs to D , . We need to introduce also thegenerator of the Ornstein-Uhlenbeck semigroup, defined as L = − ∞ X q =0 q · proj( · | C q ) , where proj( · | C q ) is the orthogonal projection operator onto the q -th Wiener chaos C q . Thedomain of L is D , , equivalently the space of Gaussian subordinated random variables F such that + ∞ X q =1 q k F [ q ] k L ( P ) < + ∞ . L is defined as L − = − P ∞ q =1 1 q · proj( · | C q ) and satisfies,for each F ∈ L ( P ), LL − F = F − E [ F ] . Stein’s method for Normal approximations and Malliavin calculus applied on Wiener chaoseslead to so-called Fourth Moment Theorems [NP09], [NP12, Chapters 5, 6]. Briefly, for asequence of centered random variables ( F n ) n ≥ living in a fixed Wiener chaos C q , suchthat Var( F n ) = 1 for all n = 1 , , . . . , convergence in law to a standard Gaussian randomvariable is equivalent to the sequence of fourth cumulants (cum ( F n )) n converging to 0[NP12, Theorem 5.2.7]. Moreover, the quantity | cum ( F n ) | gives information [NP12, The-orem 5.2.6] on the rate of convergence to the limiting distribution in various probabilitymetrics, the Wassertein distance (1.10) included. The contribution of cum ( F n ) can beexpressed in terms of contractions (3.31) of the kernel f q,n , where F n = I q ( f q,n ) (see [NP12,Lemma 5.2.4]). For a sequence of random vectors whose components lie in different chaoses,convergence to a multivariate Gaussian is equivalent to componentwise convergence to thenormal distribution [PT05].We will properly state these results for H = L ( S d ) but they hold in much more generality[NP12, Chapters 5, 6]. Here and in what follows, d T V and d K shall denote the TotalVariation and Kolmogorov distance, respectively (see [NP12, § C.2] e.g.).
Proposition 3.1 (Theorem 5.1.3 in [NP12]) . Let F ∈ D , be such that E [ F ] = 0 , E [ F ] = σ < + ∞ and Z ∼ N (0 , σ ) a centered Gaussian random variable with variance σ . Thenwe have d W ( F, Z ) ≤ r σ π E (cid:2)(cid:12)(cid:12) σ − h DF, − DL − F i H (cid:12)(cid:12)(cid:3) . Also, assuming in addition that F has a densityd T V ( F, Z ) ≤ σ E (cid:2)(cid:12)(cid:12) σ − h DF, − DL − F i H (cid:12)(cid:12)(cid:3) , d K ( F, Z ) ≤ σ E (cid:2)(cid:12)(cid:12) σ − h DF, − DL − F i H (cid:12)(cid:12)(cid:3) . In the special case where F = I q ( f ) for f ∈ L ( S d ) ⊙ q , then from [NP12, Theorem 5.2.6], E (cid:2)(cid:12)(cid:12) σ − h DF, − DL − F i H (cid:12)(cid:12)(cid:3) ≤ vuut q q − X r =1 r r ! (cid:18) qr (cid:19) (2 q − r )! k f e ⊗ r f k H ⊗ q − r . (3.33)Note that in (3.33) we can replace k f e ⊗ r f k H ⊗ q − r with the norm of the unsymmetryzedcontraction k f ⊗ r f k H ⊗ q − r , since k f e ⊗ r f k H ⊗ q − r ≤ k f ⊗ r f k H ⊗ q − r by the triangle inequality. 14 Proof of Proposition 1.2
From Lemma 2.1 we have, by the orthogonality property of chaotic projections and (3.29)Var( D ℓ ) = + ∞ X q =1 (cid:18) J q +1 (2 q + 1)! (cid:19) Z S d Z S d E [ H q +1 ( T ℓ ( x )) H q +1 ( T ℓ ( y ))] dxdy = + ∞ X q =1 J q +1 (2 q + 1)! Z S d Z S d G ℓ ; d (cos d ( x, y )) q +1 dxdy. (4.34)The isotropy property of the integrand function in the r.h.s. of (4.34) and the choice ofstandard coordinates on the hypersphere allow to writeVar( D ℓ ) = + ∞ X q =1 J q +1 (2 q + 1)! | S d || S d − | Z π G ℓ ; d (cos ϑ ) q +1 (sin ϑ ) d − dϑ, (4.35)which coincides with (2.22). We are now ready to give the proof of Proposition 1.2, inspiredby the proofs of [MW11, Proposition 4.2, Theorem 1.2]. Proof of Proposition 1.2.
Let us bear in mind (2.23). In [MR15, Proposition 2.2] ithas been proven that, as ℓ → + ∞ ,lim ℓ → + ∞ ℓ d Z π/ G ℓ ; d (cos ϑ ) q +1 (sin ϑ ) d − dϑ = c q +1; d , (4.36)where c q +1; d is given by c q +1; d := R + ∞ e J d ( ψ ) q +1 ψ d − dψ , e J d being defined as in (1.15).Therefore, as anticipated in § D ℓ ) ∼ C d ℓ d , (4.37)where the leading constant C d (uniquely depending on d ) satisfies C d = 2 | S d || S d − | + ∞ X q =1 J q +1 (2 q + 1)! c q +1; d . (4.38)Before proving (4.37) that coincides with (1.13), let us check that C d >
0, assuming (4.38)is true. Actually, the r.h.s. of (4.38) is a series of nonnegative terms and from [AAR99, p.217] we have c d = (cid:18) d − (cid:18) d − (cid:19) ! (cid:19) d − ( d − ) − √ π Γ (cid:0) d − (cid:1) > . The previous argument proves also (1.16).Moreover, assuming (4.38) is true and keeping in mind (1.15), we get C d = 2 | S d || S d − | + ∞ X q =1 J q +1 (2 q + 1)! Z + ∞ e J d ( ψ ) q +1 ψ d − dψ = 2 | S d || S d − | Z + ∞ ∞ X q =1 J q +1 (2 q + 1)! e J d ( ψ ) q +1 ψ d − dψ = 4 π | S d || S d − | Z + ∞ (cid:16) arcsin (cid:16) e J d ( ψ ) (cid:17) − e J d ( ψ ) (cid:17) ψ d − dψ. (4.39)15ctually, it is readily checked that the sequence (cid:0) J q +1 / (2 q + 1)! (cid:1) q gives the coefficientsin the Taylor expansion for the arcsin function. Actually, since H q (0) = ( − q (2 q − q − ≤ q − J q +1 (2 q + 1)! = 2 π (2 q )!4 q ( q !) (2 q + 1) . (4.40)To justify the exchange of the integration and summation order in (4.39), we consider, for m ∈ N , m >
1, the finite summation m X q =1 J q +1 (2 q + 1)! Z + ∞ e J d ( ψ ) q +1 ψ d − dψ and using the asymptotics J q +1 (2 q + 1)! ∼ cq / (4.41)for some c > § m → + ∞ .Let us now formally prove the asymptotic result for the variance (4.37). Note that, sincethe family ( G ℓ ; d ) ℓ is uniformly bounded by 1, + ∞ X q = m +1 J q +1 (2 q + 1)! Z π/ | G ℓ ; d (cos θ ) | q +1 (sin θ ) d − dθ ≤ + ∞ X q = m +1 J q +1 (2 q + 1)! Z π/ | G ℓ ; d (cos θ ) | (sin θ ) d − dθ ≪ ℓ d + ∞ X q = m +1 q / ≪ √ m ℓ d . (4.42)Therefore, for m = m ( ℓ ) to be chosenVar( D ℓ ) = 2 | S d || S d − | m X q =1 J q +1 (2 q + 1)! Z π/ G ℓ ; d (cos θ ) q +1 (sin θ ) d − dθ + O (cid:18) √ m ℓ d (cid:19) . From (4.36), we can writeVar( D ℓ ) = C d,m · ℓ d + o (cid:16) ℓ − d (cid:17) + O (cid:18) √ m ℓ d (cid:19) , where C d,m := 2 | S d || S d − | m X q =1 J q +1 (2 q + 1)! c q +1; d . Now since C d,m → C d as m → + ∞ , we can conclude. Bearing in mind (2.21), let us set for simplicity of notation h ℓ ;2 q +1 ,d := Z S d H q +1 ( T ℓ ( x )) dx. m ∈ N , m > D mℓ := m − X q =1 J q +1 (2 q + 1)! h ℓ ;2 q +1 ,d . We still have to introduce some more notation. As usual, Z ∼ N (0 , Z ℓ,m shalldenote a centered Gaussian random variable with variance σ ℓ,m := Var( D mℓ )Var( D ℓ ) . We are nowready to give the proof of the quantitative CLT for the defect on the hypersphere in anydimension. The technique we adopt was used to prove rate of convergence to the Gaussiandistribution in other circumstances (e.g. [CM15, Pha13]). Proof of Theorem 1.3.
By the triangle inequalityd W D ℓ p Var( D ℓ ) , Z ! ≤ d W D ℓ p Var( D ℓ ) , D mℓ p Var( D ℓ ) ! + d W D mℓ p Var( D ℓ ) , Z ℓ,m ! + d W ( Z ℓ,m , Z ) . Let us denote I := d W (cid:18) D ℓ √ Var( D ℓ ) , D mℓ √ Var( D ℓ ) (cid:19) , I := d W (cid:18) D mℓ √ Var( D ℓ ) , Z ℓ,m (cid:19) and I :=d W ( Z ℓ,m , Z ). Bounding the contribution of I . By definition of Wasserstein distance (1.10) we have I = d W D ℓ p Var( D ℓ ) , D mℓ p Var( D ℓ ) ! ≤ E D ℓ p Var( D ℓ ) − D mℓ p Var( D ℓ ) ! = 1Var( D ℓ ) E h ( D ℓ − D mℓ ) i = 1Var( D ℓ ) + ∞ X q = m J q +1 ((2 q + 1)!) Var ( h ℓ ;2 q +1 ,d ) , where the last equality still follows from the othogonality property of chaotic projections.From (4.42) and (1.13), one infers thatd W D ℓ p Var( D ℓ ) , D mℓ p Var( D ℓ ) ! = O (cid:18) √ m (cid:19) . (5.43) Bounding the contribution of I . We will use some technical results involving Stein-Malliavin approximation methods [NP12, Chapters 5, 6]. Let us set, from now on, H := L ( S d ).From Proposition 3.1 and (3.33) we deduced W D mℓ p Var( D ℓ ) , Z ℓ,m ! ≤ r π s Var( D ℓ )Var( D mℓ ) E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Var( D mℓ )Var( D ℓ ) − * D D mℓ p Var( D ℓ ) , − DL − D mℓ p Var( D ℓ ) + H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) E (cid:2)(cid:12)(cid:12) Var( D mℓ ) − (cid:10) DD mℓ , − DL − D mℓ (cid:11) H (cid:12)(cid:12)(cid:3) . (5.44)17ince Var( D mℓ ) = m − X q =1 J q +1 ((2 q + 1)!) Var( h ℓ ;2 q +1 ,d ) , we can write for the r.h.s. of (5.44) r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) E (cid:2)(cid:12)(cid:12) Var( D mℓ ) − (cid:10) DD mℓ , − DL − D mℓ (cid:11) H (cid:12)(cid:12)(cid:3) ≤ r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) m − X q =1 J q +1 ((2 q + 1)!) E (cid:2)(cid:12)(cid:12) Var( h ℓ ;2 q +1 ,d ) − (cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 q +1 ,d (cid:11) H (cid:12)(cid:12)(cid:3) + r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) m − X q =1 J q +1 (2 q + 1)! X q = p J p +1 (2 p + 1)! E (cid:2)(cid:12)(cid:12)(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 p +1 ,d (cid:11) H (cid:12)(cid:12)(cid:3) ≤ r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) m − X q =1 J q +1 ((2 q + 1)!) q Var (cid:0) h Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 q +1 ,d i H (cid:1) + r π s Var( D ℓ )Var( D mℓ ) 1Var( D ℓ ) m − X q =1 J q +1 (2 q + 1)! X q = p J p +1 (2 p + 1)! r E h h Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 p +1 ,d i H i , (5.45)where for the last step we used [NP12, Theorem 2.9.1] and Cauchy-Schwarz inequality.Now, Lemma 7.1 which is collected in the Appendix 7.2 and whose proof relies in particularon [MR15, Proposition 4.2, Proposition 4.3] and Lemma 1.4, allows one to write, for some C >
0, Var (cid:0)(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 q +1 ,d (cid:11) H (cid:1) ≤ C (2 q + 1) ((2 q )!) q R ℓ ; d , E h(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 p +1 ,d (cid:11) H i ≤ C (2 q + 1) (2 q )!(2 p )!3 q +2 p R ℓ ; d , where R ℓ ;2 := log ℓℓ / and for d > R ℓ ; d := 1 ℓ d +( d − / . Therefore, Lemma 7.1, (5.44) and (5.45) gived W D mℓ p Var( D ℓ ) , Z ℓ,m ! ≤ r π s Var( D ℓ )Var( D mℓ ) p R ℓ ; d Var( D ℓ ) m − X q =1 J q +1 ((2 q + 1)!) p C (2 q + 1) ((2 q )!) q + r π s Var( D ℓ )Var( D mℓ ) p R ℓ ; d Var( D ℓ ) m − X q =1 J q +1 (2 q + 1)! X q = p J p +1 (2 p + 1)! p C (2 q + 1) (2 q )!(2 p )!3 q +2 p . Now m − X q =1 J q +1 ((2 q + 1)!) p (2 q + 1) ((2 q )!) q = m − X q =1 J q +1 (2 q + 1)! 3 q m − X q =1 J q +1 ((2 q + 1)!) p (2 q + 1) ((2 q )!) q ≤ C m (5.46)and analogously m − X q =1 J q +1 (2 q + 1)! X q = p J p +1 (2 p + 1)! p (2 q + 1) (2 q )!(2 p )!3 q +2 p ≤ C m , (5.47)for some positive constant C . Finally, thanks to Proposition 1.2 and since σ ℓ ; m ≤ C − m − / (by (4.42)), we can writed W D mℓ p Var( D ℓ ) , Z ℓ,m ! ≤ C m r − m − / ℓ d p R ℓ ; d . (5.48) Bounding the contribution of I . Proposition 3.6.1 in [NP12] and (4.42) gived W ( Z ℓ,m , Z ) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) Var( D ℓ − D mℓ )Var( D ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ √ m . (5.49) Optimazing on m . Summing up the three bounds (5.43), (5.48) and (5.49) with thechoice of the speed m ≍ log ℓ α , for some α ∈ (0 , / We shall use sometimes in this section the shorthand notation X m := n ℓ ; d X m =1 and X m ,m := n ℓ ; d X m ,m =1 . Moreover we will drop the dependence of d in (1.18), for brevity. Proof of Lemma 1.4 assuming Lemma 1.5.
Let us study the following multipleintegral which gives the major contribution to the 4-th cumulant of R S d H ( T ℓ ( x )) dx , asalready stated in § Z ( S d ) G ℓ ; d (cos d ( w, z )) G ℓ ; d (cos d ( w, w ′ )) G ℓ ; d (cos d ( w ′ , z ′ )) G ℓ ; d (cos d ( z ′ , z )) dwdzdw ′ dz ′ . The addition formula for Gegenbauer polynomials (1.6) allows one to write Z S d G ℓ ; d (cos d ( w, z )) G ℓ ; d (cos d ( w, w ′ )) dw = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ′ ,m ′ =1 Y ℓ,m ( z ) Y ℓ,m ′ ( w ′ ) Y ℓ,m ′ ( w ′ ) Z S d Y ℓ,m ( w ) Y ℓ,m ′ ( w ) Y ℓ,m ′ ( w ) dw = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ′ ,m ′ =1 Y ℓ,m ( z ) Y ℓ,m ′ ( w ′ ) Y ℓ,m ′ ( w ′ ) G ℓ,m ℓ,m ′ ,ℓ,m ′ (6.50)19here G ℓ,m ℓ,m ′ ,ℓ,m ′ has been defined in (1.18). The same argument applied in order to deduce(6.50) gives Z S d G ℓ ; d (cos d ( w ′ , z ′ )) G ℓ ; d (cos d ( z ′ , z )) dz ′ = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ′ ,m ,m =1 Y ℓ,m ′ ( w ′ ) Y ℓ,m ( z ) Y ℓ,m ( z ) G ℓ,m ′ ℓ,m ,ℓ,m . (6.51)We are thus left with Z ( S d ) G ℓ ; d (cos d ( w, z )) G ℓ ; d (cos d ( w, w ′ )) G ℓ ; d (cos d ( w ′ , z ′ )) G ℓ ; d (cos d ( z ′ , z )) dwdzdw ′ dz ′ = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ,m ,m ′ ,m ′ ,m ′ =1 G ℓ,m ℓ,m ′ ,ℓ,m ′ G ℓ,m ′ ℓ,m ,ℓ,m ×× Z ( S d ) Y ℓ,m ′ ( w ′ ) Y ℓ,m ( z ) Y ℓ,m ( z ) Y ℓ,m ( z ) Y ℓ,m ′ ( w ′ ) Y ℓ,m ′ ( w ′ ) dzdw ′ = (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ,m ,m ′ ,m ′ ,m ′ =1 G ℓ,m ℓ,m ′ ,ℓ,m ′ G ℓ,m ′ ℓ,m ,ℓ,m G ℓ,m ℓ,m ,ℓ,m G ℓ,m ′ ℓ,m ′ ,ℓ,m ′ . (6.52)Lemma 1.5 states that X m ,m G ℓ,m ′ ℓ,m ,ℓ,m G ℓ,m ℓ,m ,ℓ,m = g ℓ ; d δ m ′ m , (6.53)where g ℓ ; d := ( n ℓ ; d ) | S d | | S d − || S d | Z − G ℓ ; d ( t ) (cid:16)p − t (cid:17) d − dt. Plugging (6.53) into (6.52) and applying once more Lemma 1.5 we find (cid:18) | S d | n ℓ ; d (cid:19) n ℓ ; d X m ,m ,m ,m ′ ,m ′ ,m ′ =1 G ℓ,m ℓ,m ′ ,ℓ,m ′ G ℓ,m ′ ℓ,m ,ℓ,m G ℓ,m ℓ,m ,ℓ,m G ℓ,m ′ ℓ,m ′ ,ℓ,m ′ = (cid:18) | S d | n ℓ ; d (cid:19) g ℓ ; d X m ,m ′ ,m ′ G ℓ,m ℓ,m ′ ,ℓ,m ′ G ℓ,m ℓ,m ′ ,ℓ,m ′ = (cid:18) | S d | n ℓ ; d (cid:19) g ℓ ; d X m X m ′ ,m ′ G ℓ,m ℓ,m ′ ,ℓ,m ′ G ℓ,m ℓ,m ′ ,ℓ,m ′ | {z } = g ℓ ; d = (cid:18) | S d | n ℓ ; d (cid:19) ( g ℓ ; d ) n ℓ ; d = | S d | (cid:18) n ℓ ; d (cid:19) ( g ℓ ; d ) . (6.54)From [MR15, Proposition 2.2], it is readily checked that g ℓ ; d ≍ ℓ d − . Hence (6.54) givescum (cid:18)Z S d H ( T ℓ ( x )) dx (cid:19) ≪ ℓ d − D R S d H ( T ℓ ( x )) dx q Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) , Z = O vuut cum (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) Var (cid:0)R S d H ( T ℓ ( x )) dx (cid:1) = O r ℓ d − ! . Proof.
Since by definition G ℓ,Mℓ,m ,ℓ,m = Z S d Y ℓ,m ( x ) Y ℓ,m ( x ) Y ℓ,M ( x ) dx, we can write X m ,m G ℓ,Mℓ,m ,ℓ,m G ℓ,M ′ ℓ,m ,ℓ,m = X m ,m Z S d Y ℓ,m ( x ) Y ℓ,m ( x ) Y ℓ,M ( x ) dx Z S d Y ℓ,m ( y ) Y ℓ,m ( y ) Y ℓ,M ′ ( y ) dy = Z S d Z S d dxdyY ℓ,M ( x ) Y ℓ,M ′ ( y ) X m Y ℓ,m ( x ) Y ℓ,m ( y ) X m Y ℓ,m ( x ) Y ℓ,m ( y )= Z S d Z S d dxdyY ℓ,M ( x ) Y ℓ,M ′ ( y ) (cid:18) n ℓ ; d | S d | (cid:19) G ℓ ; d (cos d ( x, y )) , (6.55)where in the last equality we used twice the addition formula (1.6) for Gegenabuer poly-nomials. The family (cid:16)p | S d − | n ℓ ; d / | S d | G ℓ ; d (cid:17) ℓ being orthonormal on [ − , G ℓ ; d ( t ) = ℓ X j =0 γ j G j ; d ( t ) , t ∈ [ − , , (6.56)where the coefficients γ j := γ j ( ℓ ; d ) are given by γ j = n ℓ ; d | S d − || S d | Z − G ℓ ; d ( t ) G j ; d ( t )( p − t ) d − dt. Therefore, plugging (6.56) into (6.55), by the orthormality property of hyperspherical har-monics one deduces X m ,m G ℓ,Mℓ,m ,ℓ,m G ℓ,M ′ ℓ,m ,ℓ,m = Z S d Z S d dxdyY ℓ,M ( x ) Y ℓ,M ′ ( y ) (cid:18) n ℓ ; d | S d | (cid:19) X j γ j G j ; d (cos d ( x, y ))= Z S d Z S d dxdyY ℓ,M ( x ) Y ℓ,M ′ ( y ) (cid:18) n ℓ ; d | S d | (cid:19) X j γ j X k | S d | n j ; d Y j,k ( x ) Y j,k ( y )= (cid:18) n ℓ ; d | S d | (cid:19) X j γ j | S d | n j ; d X k Z S d Y ℓ,M ( x ) Y j,k ( x ) dx | {z } = δ jℓ δ kM Z S d Y ℓ,M ′ ( y ) Y j,k ( y ) dy | {z } = δ jℓ δ kM ′ = n ℓ ; d | S d | γ ℓ δ M ′ M . This concludes the proof. 21
Appendix
Proof.
By (1.7), one deduces that | D ℓ | ≤ | S d | a.s. and hence D ℓ ∈ L ( P ). Recall that wecan write D ℓ = 2 Z S d (0 , + ∞ ) ( T ℓ ( x )) dx − | S d | . The chaotic expansion § (0 , + ∞ ) is given by (see e.g. [MW14]and the references therein)1 (0 , + ∞ ) ( · ) = 12 + X q ≥ φ (0) H q (0)(2 q + 1)! H q +1 ( · ) , where φ still denotes the p.d.f. of the standard Gaussian law and ( H k ) k ≥ the sequence ofHermite polynomials. Hence in particular, X q ≥ ( φ (0) H q (0)) (2 q + 1)! = Φ(0)(1 − Φ(0)) < + ∞ , (7.57)Φ still denoting the cumulative distribution function of a standard Gaussian random vari-able. Actually, it is easy to check that, for Z ∼ N (0 , E [1 (0 , + ∞ ) ( Z )] = 1 /
2, whereas for k ≥ E [1 (0 , + ∞ ) ( Z ) H k ( Z )] = Z + ∞ ( − k φ − ( t ) d k φdt k ( t ) φ ( t ) dt = ( − k d k − φdt k − ( t ) (cid:12)(cid:12)(cid:12) + ∞ = − φ ( t ) H k − ( t ) (cid:12)(cid:12)(cid:12) + ∞ = φ (0) H k − (0) , which vanishes if k is even. For m ∈ N , m >
0, let us consider the random variable U mℓ := 2 Z S d
12 + m X q =0 φ (0) H q (0)(2 q + 1)! H q +1 ( T ℓ ( x )) dx − | S d | = m X q =1 φ (0) H q (0)(2 q + 1)! Z S d H q +1 ( T ℓ ( x )) dx, where the sum starts from q = 1 since H ( t ) = t and hyperspherical harmonics have zeromean over S d . Let us set moreover J q +1 := 2 φ (0) H q (0) . In what follows, we shall show that the sequence of random variables ( U mℓ ) m is a Cauchysequence in L ( P ). By the orthogonality property of chaotic projections and (3.29), we havefor m, n ∈ N , n, m > E [( U mℓ − U m + nℓ ) ] = m + n X q = m +1 J q +1 (2 q + 1)! Z ( S d ) G ℓ ; d (cos d ( x, y )) q +1 dxdy. E [( U mℓ − U m + nℓ ) ] ≤ | S d | m + n X q = m +1 J q +1 (2 q + 1)! ;hence (7.57) allows to conclude the proof. Let us denote H := L ( S d ). Lemma 7.1.
There exists
C > such that for integers q, p ≥ , q ≤ p , Var (cid:0)(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 q +1 ,d (cid:11) H (cid:1) ≤ C (2 q + 1) ((2 q )!) q R ℓ ; d , E h(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 p +1 ,d (cid:11) H i ≤ C (2 q + 1) (2 q )!(2 p )!3 q +2 p R ℓ ; d , (7.58) where R ℓ ;2 := log ℓℓ / and for d > R ℓ ; d := 1 ℓ d +( d − / . Proof.
Recall that h ℓ ;2 q +1 ,d can be expressed as a multiple Wiener-Itˆo integral of order q (see § h ℓ ;2 q +1 ,d L = Z ( S d ) q g ℓ ;2 q +1 ,d ( y , y , . . . , y q ) dW ( y ) dW ( y ) . . . dW ( y q ) =: I q ( g ℓ ;2 q +1 ,d ) , where the function g ℓ ;2 q +1 ,d is given by g ℓ ;2 q +1 ,d ( y , y , . . . , y q ) := Z S d (cid:18) n ℓ ; d | S d | (cid:19) q/ G ℓ ; d (cos d ( x, y )) · · · G ℓ ; d (cos d ( x, y q )) dx. Similar arguments as those in the proof of [CM15, Lemma 6.1] allow one to have, for integers p, q ≥
1, the following new estimatesVar (cid:0)(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 q +1 ,d (cid:11) H (cid:1) ≤ (2 q + 1) q X r =1 (( r − (cid:18) qr − (cid:19) (2(2 q + 1) − r )! k g ℓ ;2 q +1 ,d ⊗ r g ℓ ;2 q +1 ,d k H ⊗ q +1) − r , (7.59)and moreover for q ≤ p E h(cid:10) Dh ℓ ;2 q +1 ,d , − DL − h ℓ ;2 p +1 ,d (cid:11) H i = (2 q + 1) q +1 X r =1 (( r − (cid:18) qr − (cid:19) (cid:18) pr − (cid:19) (2 q + 2 p + 2 − r )! k g ℓ ;2 q +1 ,d e ⊗ r g ℓ ;2 p +1 ,d k H ⊗ n ≤ (2 q + 1) q +1 X r =1 (( r − (cid:18) qr − (cid:19) (cid:18) pr − (cid:19) (2 q + 2 p + 2 − r )! k g ℓ ;2 q +1 ,d ⊗ r g ℓ ;2 p +1 ,d k H ⊗ n , (7.60)23here n := 2 q + 2 p + 2 − r for notational simplicity. Now from [MR15, Proposition 4.1]we know the explicit formula for the norm of contractions: for q ≤ p k g ℓ ;2 q +1 ,d ⊗ r g ℓ ;2 p +1 ,d k H ⊗ n = Z ( S d ) G ℓ ; d (cos d ( x , x )) r G ℓ ; d (cos d ( x , x )) q +1 − r ×× G ℓ ; d (cos d ( x , x )) r G ℓ ; d (cos d ( x , x )) q +1 − r dx, where dx := dx dx dx dx . Thanks to [MR15, Proposition 4.2, Proposition 4.3] (for q ≥ q = 1) we have, as ℓ → + ∞ , k g ℓ ;2 q +1 , ⊗ r g ℓ ;2 p +1 , k H ⊗ n = O ( R ℓ ; d ) , (7.61)where R ℓ ;2 = log ℓ/ℓ / and R ℓ ; d = 1 /ℓ d +( d − / for d >
2. Note that O ’ notation isindependent of q and p .As stated in [CM15, (6.1),(6.2)], the following inequalities hold q X r =1 (( r − (cid:18) qr − (cid:19) (2(2 q + 1) − r )! ≤ ((2 q )!) q , q +1 X r =1 (( r − (cid:18) qr − (cid:19) (cid:18) pr − (cid:19) (2 q + 2 p + 2 − r )! ≤ (2 q )!(2 p )!3 q +2 p . (7.62)Plugging (7.62) and (7.61) into (7.59) and (7.60), one infers (7.58). References [AT07] R. J. Adler and J. E. Taylor.
Random fields and geometry . Springer Monographsin Mathematics. Springer, New York, 2007.[AAR99] G. E. Andrews, R. Askey, and R. Roy.
Special functions , volume 71 of
Encyclopediaof Mathematics and its Applications . Cambridge University Press, Cambridge, 1999.[BMW17] J. Benatar, D. Marinucci, and I. Wigman. Planck-scale distribution of nodallength of arithmetic random waves. arXiv:1710.06153 .[Ber77] M. V. Berry. Regular and irregular semiclassical wavefunctions.
Journal of PhysicsA: Mathematical and Theoretical 10 , 12:2083–2091, 1977.[Ber02] M. V. Berry. Statistics of nodal lines and points in chaotic quantum billiards:perimeter corrections, fluctuations, curvature.
Journal of Physics. A. Mathematicaland General , 35(13):3025–3038, 2002.[BG72] J. Br¨uning and D. Gromes. ¨Uber die L¨ange der Knotenlinien schwingender Mem-branen.
Mathematische Zeitschrift , 124:79–82, 1972.[Br¨u78] J. Br¨uning. ¨Uber Knoten von Eigenfunktionen des Laplace-Beltrami-Operators.
Mathematische Zeitschrift , 158(1):15–21, 1978.[CM15] V. Cammarota and D. Marinucci. On the limiting behaviour of needlets polyspec-tra.
Annales de l’Institut Henri Poincar´e Probabilit´es et Statistiques , 51(3):1159–1189,2015. 24CM16] V. Cammarota and D. Marinucci. A quantitative central limit theorem for theEuler-Poincar´e characteristic of random spherical eigenfunctions.
The Annals of Prob-ability , in press.[CMW16] V. Cammarota, D. Marinucci, and I. Wigman. Fluctuations of the Euler-Poincar´echaracteristic for random spherical harmonics.
Proceedings of the American Mathemat-ical Society , 144(11):4759–4775, 2016.[DF88] H. Donnelly and C. Fefferman. Nodal sets of eigenfunctions on Riemannian mani-folds.
Inventiones Mathematicae , 93(1):161–183, 1988.[Dur16] C. Durastanti. Adaptive global thresholding on the sphere.
Journal of MultivariateAnalysis , 151:110–132, 2016.[Far08] J. Faraut.
Analysis on Lie groups , volume 110 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 2008. An introduction.[GRS13] A. Ghosh, A. Reznikov, and P. Sarnak. Nodal domains of Maass forms I.
Geometricand Functional Analysis , 23(5):1515–1568, 2013.[JZ16] J. Jung and S. Zelditch. Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary.
Mathematische Annalen , 364(3-4):813–840, 2016.[KKW13] M. Krishnapur, P. Kurlberg, and I. Wigman. Nodal length fluctuations for arith-metic random waves.
Annals of Mathematics (2) , 177(2):699–737, 2013.[Mar06] D. Marinucci. High-resolution asymptotics for the angular bispectrum of sphericalrandom fields.
The Annals of Statistics , 34(1):1–41, 2006.[Mar08] D. Marinucci. A central limit theorem and higher order results for the angularbispectrum.
Probability Theory and Related Fields , 141(3-4):389–409, 2008.[MP11] D. Marinucci and G. Peccati.
Random fields on the sphere , volume 389 of
LondonMathematical Society Lecture Note Series . Cambridge University Press, Cambridge,2011.[MPRW16] D. Marinucci, G. Peccati, M. Rossi, and I. Wigman. Non-Universality of nodallengths distribution for arithmetic random waves.
Geometric and Functional Analysis ,26(3):926–960, 2016.[MR15] D. Marinucci and M. Rossi. Stein-Malliavin approximations for nonlinear function-als of random eigenfunctions on S d . Journal of Functional Analysis , 268(8):2379–2420,2015.[MRW17] D. Marinucci, M. Rossi, and I. Wigman. The asymptotic equivalence of the sam-ple trispectrum and the nodal length for random spherical harmonics. arXiv:1705.05747 [MW11] D. Marinucci and I. Wigman. The defect variance of random spherical harmonics.
Journal of Physics A: Mathematical and Theoretical , 44:355206, 2011.[MW14] D. Marinucci and I. Wigman. On nonlinear functionals of random spherical eigen-functions.
Communications in Mathematical Physics , 327(3):849–872, 2014.25Mec09] E. Meckes. On the approximate normality of eigenfunctions of the Laplacian.
Transactions of the American Mathematical Society , 361(10):5377–5399, 2009.[NS09] F. Nazarov and M. Sodin. On the number of nodal domains of random sphericalharmonics.
American Journal of Mathematics , 131(5):1337–1357, 2009.[NP09] I. Nourdin and G. Peccati. Stein’s method on Wiener chaos.
Probability Theoryand Related Fields , 145(1-2):75–118, 2009.[NP12] I. Nourdin and G. Peccati.
Normal approximations with Malliavin calculus , volume192 of
Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge,2012.[PT05] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochasticintegrals. In
S´eminaire de Probabilit´es XXXVIII , volume 1857 of
Lecture Notes inMath. , pages 247–262. Springer, Berlin, 2005.[PT11] G. Peccati and M. S. Taqqu.
Wiener chaos: moments, cumulants and diagrams ,volume 1 of
Bocconi & Springer Series . Springer, Milan; Bocconi University Press,Milan, 2011.[Pha13] V.-H. Pham. On the rate of convergence for central limit theorems of sojourn timesof Gaussian fields.
Stochastic Processes and their Applications , 123(6):2158–2174, 2013.[Ros15] M. Rossi.
The Geometry of Spherical Random fields . PhD Thesis(arXiv:1603.07575). University of Rome Tor Vergata, 2015.[RW08] Z. Rudnick and I. Wigman. On the volume of nodal sets for eigenfunctions of theLaplacian on the torus.
Annales Henri Poincar´e , 9(1):109–130, 2008.[Sze75] G. Szeg˝o.
Orthogonal polynomials . American Mathematical Society, Providence,R.I., fourth edition, 1975. American Mathematical Society, Colloquium Publications,Vol. XXIII.[VK93] N. Ja. Vilenkin and A. U. Klimyk.
Representation of Lie groups and special func-tions. Vol. 2 , volume 74 of
Mathematics and its Applications (Soviet Series) . KluwerAcademic Publishers Group, Dordrecht, 1993.[VMK88] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonski˘ı.
Quantum theory ofangular momentum . World Scientific Publishing Co., Inc., Teaneck, NJ, 1988.[Wig10] I. Wigman. Fluctuations of the nodal length of random spherical harmonics.
Com-munications in Mathematical Physics , 298(3):787–831, 2010.[Wig12] I. Wigman. On the nodal lines of random and deterministic Laplace eigenfunc-tions. In
Spectral geometry , volume 84 of
Proceedings of the International Conferenceon Spectral Geometry, Dartmouth College , pages 285–297. American Mathematical So-ciety, Providence, RI, 2012.[Yau82] S.-T. Yau. Survey on partial differential equations in differential geometry. In
Seminar on Differential Geometry , volume 102 of