Random Nodal Lengths and Wiener Chaos
Maurizia Rossi































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Maurizia Rossi





MAP5-UMR CNRS 8145, Université Paris Descartes, France 

E-mail address: 









In this survey we collect some of the recent results on the “nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics).  We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. 
Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains). 



Keywords and Phrases: nodal length, chaos expansion, random eigenfunctions, limit theorems.



AMS Classification: 60G60, 60B10, 60D05, 35P20, 58J50





§ INTRODUCTION




 §.§ Background and motivations


At the end of the 18th century, the musician and physicist
Chladni noticed that  sounds of different
pitch could be made by exciting a metal plate with the bow of a violin, depending on where the bow touched the plate. The latter was
fixed only in the center, and when there was some sand on it, for
each pitch a curious pattern appeared (see Figure <ref>).
About 60 years later, Kirchhoff, inspired also by previous contributions of Germain, Lagrange and Poisson, showed that Chladni figures on a plate correspond to the zeros of eigenpairs
(eigenvalues and corresponding eigenfunctions) of the biharmonic operator with free
boundary conditions. Nowadays nodal patterns arise in several areas, from  the musical instruments industry to the study of natural phemomena such as earthquakes. 
See <cit.> and the references therein for more details. 


Now let (ℳ,g) denote a compact smooth Riemannian manifold of dimension 2 and f:ℳ→ℝ a function on ℳ. The set f^-1(0) := { x∈ℳ : f(x) = 0} is usually called nodal set of f. 
We are interested in the case where f is an eigenfunction of the  Laplace-Beltrami operator 
 Δ on the manifold. 
There exists an orthonormal basis { f_j }_j=0^+∞ of L^2(ℳ) consisting of eigenfunctions for -Δ whose corresponding sequence of  eigenvalues {λ_j^2 }_j=0^+∞

    -Δ f_j = λ^2_j f_j
 
is non-decreasing (in particular 0=λ_0 < λ_1 ≤λ_2 ≤…). 

The nodal set of f_j  is, generically, a smooth curve (called nodal line) whose components are homeomorphic to the circle. We are interested in the geometry of f_j^-1(0) for high energy levels, i.e. as λ_j→ +∞.  

Yau's conjecture <cit.> concerns, in particular[Yau stated its conjecture in any dimension], the length of nodal lines f_j^-1(0):  there exist two positive constant c_ℳ, C_ℳ such that 

    c_ℳλ_j ≤length(f_j^-1(0)) ≤ C_ℳλ_j,

for every j≥ 1. Yau's conjecture was proved by Donnelly and Fefferman in <cit.> for real analytic manifolds, and the lower bound
was established by Logunov and Malinnikova <cit.> for the general case. 

In <cit.> Berry conjectured that the local behavior of high energy eigenfunctions f_j in eq for generic chaotic surfaces is universal. He meant that it is comparable to the behavior of the centred Gaussian field B_λ_j = { B_λ_j(x)}_x ∈^2 on the Euclidean plane whose covariance kernel is, for x,y∈^2, 

    (B_λ_j(x), B_λ_j(y))=J_0(λ_j  x - y),

J_0 being the Bessel function of order zero <cit.> and · denoting the Euclidean norm (the latter model is known as Berry's Random Wave Model). In particular, nodal lines of B_λ_j should model nodal lines of deterministic eigenfunctions f_j for large eigenvalues, allowing the investigation of associated local quantities like the (nodal) length. 
See <cit.> and the references therein for more details.



 §.§ Short plan of the survey


In the case of surfaces (ℳ, g)  with spectral degeneracies, like the two dimensional standard flat torus or the unit round sphere, it is possible to construct a random model by endowing each eigenspace with a Gaussian measure. The asymptotic behavior, in the high energy limit, of the corresponding nodal length is the subject of <ref>. In particular, two comments are in order: the asymptotic variance is smaller than expected in both cases (Berry's cancellation phenomenon), and the nature of second order fluctuations is Gaussian for the spherical case but not on the torus. In <ref> we give some insight on both these phenomena in terms of chaotic projections, notion introduced in <ref>. 

For a generic Riemannian metric on a smooth compact surface, the eigenvalues  of the Laplace-Beltrami operator are simple and one cannot define a meaningful random model as for the toral or the spherical case, that is, a single frequency random model. In view of this, in <ref> 
we work with the so-called monochromatic random waves i.e., Gaussian random linear combination of eigenfunctions  with frequencies from a short interval. Their scaling limit is Berry's RWM cov_berry, and for the latter we present some recent results on the asymptotic behavior for its nodal length, which turns out to be Gaussian as for the spherical case. 



 §.§ Acknowledgements
 The author would like to thank the organizers of the conference Probabilistic methods in spectral geometry and PDE  and the CRM Montréal for such a wonderful meeting. 
Some of the results presented in this short survey have been obtained in collaboration with Domenico Marinucci, Ivan Nourdin, Giovanni Peccati and Igor Wigman to whom the author is grateful; in particular, the author would like to thank Domenico Marinucci for the insight in <ref> and Igor Wigman for fruitful discussions we had in Montréal. Moreover, the author thanks Valentina Cammarota for a number of useful discussions, and an anonymous referee for his/her valuable comments. 
The author was financially supported by the grant F1R-MTH-PUL-15STAR
(STARS) at the University of Luxembourg and the ERC grant no. 277742 Pascal.
Her research is currently supported  by the 
Fondation Sciences Mathématiques de Paris and the project ANR-17-CE40-0008 Unirandom.



§ RANDOM NODAL LENGTHS




 §.§ The toral case




  §.§.§ Arithmetic random waves
 
The Laplacian eigenvalues for the two-dimensional standard flat torus 𝕋^2:=^2/^2 are of the form 4π^2n, where n  is an integer expressible as a sum of two squares
(n∈ S := { n = a^2 + b^2 : a,b∈}).
For n∈ S, we define Λ_n to be the set of frequencies 

    Λ_n := {ξ∈ℤ^2 : ξ= √(n)}

and we denote by 𝒩_n its cardinality (𝒩_n is the multiplicity of  eigenvalue 4π^2n). The set Λ_n in Lambda induces a probability measure μ_n  on the unit circle 𝕊^1⊂^2

    μ_n := 1/𝒩_n∑_ξ∈Λ_nδ_ξ/√(n),
 
δ_z denoting the Dirac mass at z∈^2. For more details see <cit.>.

For n∈ S, the arithmetic random wave T_n (of order n) is the Gaussian random eigenfunction on the torus defined as follows <cit.>:

    T_n(x) := 1/√(𝒩_n)∑_ξ∈Λ_na_ξ^i2π⟨ξ, x⟩,     x∈𝕋^2,

where { a_ξ}_ξ∈Λ_n is a family of i.d. standard complex Gaussian random variables[defined on some probability space (Ω, ℱ, ℙ)], and independent except for the relation a_ξ = a_-ξ that ensures T_n to be real. Recall that {^i2π⟨ξ, ·⟩}_ξ∈Λ_n is an orthonormal basis for the eigenspace related to the eigenvalue 4π^2 n. 
Equivalently, it can be defined as the centered Gaussian field on 𝕋^2 whose covariance kernel  is, for x,y∈𝕋^2,

    (T_n(x), T_n(y)) = 1/𝒩_n∑_ξ∈Λ_n^i 2π⟨ξ, x-y⟩.

There exists a density one subsequence { n_j}_j⊂ S of energy levels such that, as j→ +∞, 

    μ_n_j⇒ dθ/2π,

dθ denoting the uniform measure on 𝕊^1 (see <cit.>). 
From cov_int we have, for x,y∈𝕋^2, as j→ +∞, 

    (T_n_j(x/2π√(n_j)), T_n_j(y/2π√(n_j))) = ∫_𝕊^1^i ⟨θ, x-y⟩ dμ_n_j(θ) → J_0(x-y),

J_0 still denoting the Bessel function of order zero (cf. cov_berry).
 There exist other weak-⋆ partial limits of the sequence {μ_n}_n∈ S, partially classified in <cit.>. 



  §.§.§ Nodal lengths: some recent results


The nodal set T_n^-1(0) is a smooth curve a.s.; let us set ℒ_n := length(T_n^-1(0)). By means of Kac-Rice formulas, the expected nodal length was computed by Rudnick and Wigman in <cit.>

    [ℒ_n] = 1/2√(2)√(4π^2n),

and in <cit.> the exact asymptotic variance was found: as 𝒩_n→ +∞, 

    (ℒ_n) ∼1+μ_n(4)^2/512 4π^2n/𝒩_n^2,

μ_n(4) denoting the fourth Fourier coefficients of μ_n. In order to have an asymptotic law for the variance, it suffices to choose a subsequence { n_j}_j⊂ S  such that as j→ +∞ we have (a) 𝒩_n_j→ +∞ and (b) |μ_n_j(4)|→η,  for some η∈ [0,1]. Note that for each η∈ [0,1], there exists a subsequence { n_j} such that both (a) and (b) hold (see <cit.>). 
The second order fluctuations of ℒ_n were investigated and fully resolved in <cit.>, as follows.  

For { n_j}_j⊂ S  such that, as j→ +∞, 𝒩_n_j→ +∞ and |μ_n_j(4)|→η  for some η∈ [0,1], we have 

    ℒ_n_j - [ℒ_n_j]/√((ℒ_n_j))→^d 1/2√(1+η^2) (2 - (1-η) Z_1^2 - (1+η)Z_2^2),

where Z_1 and Z_2 are i.i.d. standard Gaussian random variables. 

A quantitative version (in Wasserstein distance) of NCLT is given in <cit.>, where an alternative proof for varTorus is given by means of chaotic expansions.  In the recent paper <cit.> the authors study the nodal length of arithmetic random waves
restricted to decreasing domains (shrinking radius-s balls)  all the way down to Planck scale (i.e., for s> n^-1/2+ε, ε >0). Remarkably, they prove that the latter is asymptotically fully correlated with ℒ_n. In <cit.> the interesection number of nodal lines corresponding to two independent arithmetic random waves with the same eigenvalue is studied, whereas 
the papers <cit.> investigate the intersection number of the nodal lines T_n^-1(0) and a fixed reference curve with nowhere zero curvature. The case of a straight line segment is studied in <cit.> by Maffucci.  There are results also in the three dimensional setting <cit.>. 





 §.§ The spherical case




  §.§.§ Random spherical harmonics

  The Laplacian eigenvalues on the two-dimensional unit round sphere 𝕊^2 are of the form ℓ(ℓ+1), where ℓ∈ℕ, and the multiplicity of the ℓ-th eigenvalue is 2ℓ+1. The ℓ-th random spherical harmonic on 𝕊^2 is the Gaussian random eigenfunction T_ℓ (abusing notation) defined as follows (see e.g. <cit.>):

    T_ℓ(x) := √(4π/2ℓ+1)∑_m=-ℓ^ℓ a_ℓ,m Y_ℓ,m(x),     x∈𝕊^2,

where { a_ℓ,m}_m=-ℓ, …, ℓ is a family of i.d. standard complex Gaussian random variables, and independent except for the relation a_ℓ,m = (-1)^ℓ a_ℓ,-m that ensures T_ℓ to be real. The family { Y_ℓ,m}_m=-ℓ,… , ℓ of spherical harmonics <cit.> is an orthonormal basis for the eigenspace related to the eigenvalue ℓ(ℓ+1). 
Equivalently, it can be defined as the centered Gaussian field on the sphere whose covariance kernel is, for x,y∈𝕊^2,

    (T_ℓ(x)), T_ℓ(y)) = P_ℓ(cos d(x,y)),

where P_ℓ denotes the ℓ-th Legendre polynomial <cit.> and d(x,y) the geodesic distance between the two points x and y. 
Hilb's asymptotic formula <cit.> states that, uniformly for  θ∈ [0, π - ε] (ε >0), as ℓ→ +∞, 

    P_ℓ(cosθ) ∼√(θ/sinθ) J_0((ℓ+1/2)θ),

cf. cov_berry.



  §.§.§ Nodal lengths: some recent results

The nodal set T_ℓ^-1(0) := { x∈𝕊^2 : T_ℓ(x) = 0} is a smooth curve a.s.
The mean of the nodal length (abusing notation) ℒ_ℓ := length(T_ℓ^-1(0)) was computed in <cit.> 

    [ℒ_ℓ] = 4π·1/2√(2)√(ℓ(ℓ+1));

note that the coefficient 1/2√(2) in mean_sphere is universal (cf. mean_torus). 
The asymptotic behaviour of the variance was given in <cit.>: as ℓ→ +∞, 

    (ℒ_ℓ) ∼1/32logℓ.

The second order fluctuations of ℒ_ℓ have been investigated and fully resolved in <cit.>:

As ℓ→ +∞, 

    ℒ_ℓ - [ℒ_ℓ]/√((ℒ_ℓ))→^d Z,

where Z is a standard Gaussian random variable. 




 §.§ Some comments






(i) The mean of the nodal length in both the toral mean_torus and the spherical case mean_sphere is of the form 1/2√(2) (a universal constant) times the square root of the corresponding eigenvalue (cf yau_conj) times the area of the surface.

(ii) The order of magnitude of the asymptotic variance of the nodal length in both the toral  varTorus and the spherical setting varSphere is smaller than expected. Indeed, Kac-Rice formula <cit.> suggests that its variance is asymptotically proportional to the corresponding eigenvalue times the second moment of the covariance kernel which is, up to a constant factor, the inverse of the eigenvalue's multiplicity (n/𝒩_n on 𝕋^2 and ℓ for 𝕊^2).  This fact is known as Berry's cancellation phenomenon, indeed it was predicted by Berry in <cit.> and proved by Wigman in <cit.> on the sphere and in <cit.> on the torus. 

(iii) Theorem <ref> reveals in particular the non-Gaussian asymptotic nature of the toral nodal length, in contrast with the Gaussian fluctuations for the spherical case (Theorem <ref>). 

We will give some insights for this remark in <ref>. 




§ CHAOTIC EXPANSIONS


Before briefly introducing the notion of Wiener chaos, let us recall an integral representation for the nodal length (here we state it for the spherical case but it is – essentially – valid in general, thanks to the coarea formula <cit.>). 

The nodal length on the sphere can be (formally) written as 

    ℒ_ℓ = ∫_𝕊^2δ_0(T_ℓ(x)) ∇ T_ℓ(x) dx

where δ_0 denotes the Dirac mass in 0, ∇ T_ℓ the gradient field and · the Euclidean norm in ^2. 


Indeed, let us consider the -approximating random variable

    ℒ_ℓ^ := 1/2∫_𝕊^2 1_[-, ](T_ℓ(x)) ∇ T_ℓ(x)  dx,

where 1_[-, ] denotes the indicator function of the interval [-, ]. It is possible to prove that 

    lim_→ 0ℒ_ℓ^ = length(T_ℓ^-1(0)),

both a.s. and in L^2(), see <cit.>, thus justifying rapSphere. 
In particular, ℒ_ℓ is a square integrable functional of a Gaussian field; this is the key point for the theory of chaotic expansions to apply. 

Analogously, for the nodal length of arithmetic random waves we have (with obvious notation)

    ℒ_n = ∫_𝕋^2δ_0(T_n(x)) ∇ T_n(x) dx.

See <cit.> for details. 



 §.§ Wiener chaos
 

In this part we introduce the notion of Wiener chaos restricting ourselves to our specific spherical setting (the toral case is analogous). For a complete discussion see <cit.> and the references therein. 

Denote by {H_k}_k≥ 0 the sequence of Hermite polynomials on ℝ; these polynomials are defined recursively as follows: H_0 ≡ 1 and
 
    H_k(t) = tH_k-1(t) - H'_k-1(t),      k≥ 1.

Recall that ℍ := {(k!)^-1/2 H_k, k≥ 0} constitutes a complete orthonormal system in the space of square integrable real functions L^2(γ) w.r.t. the standard Gaussian density γ on the real line.

Random spherical harmonics (<ref>) are a by-product of a family of complex-valued Gaussian random variables {a_ℓ,m : ℓ = 0, 1, 2, …, m=-ℓ, …, ℓ} such that (a) every a_ℓ,m has the form x_ℓ,m+iy_ℓ,m, where x_ℓ,m and y_ℓ,m are two independent real-valued Gaussian random variables with mean zero and variance 1/2; (b) a_ℓ,m and a_ℓ',m' are  independent whenever ℓℓ' or m'∉{m,-m}, and (c) a_ℓ,m =(-1)^ℓ a_ℓ, -m. Define the space A to be the closure in L^2(ℙ) of all real finite linear combinations of random variables ξ of the form 
    ξ = z   a_ℓ,m + z  (-1)^ℓ a_ℓ, -m,     z∈ℂ,
 thus A is a real centered Gaussian Hilbert subspace
of L^2(ℙ).

Let us fix now an integer q≥ 0; the q-th Wiener chaos C_q associated with A is defined as the closure in L^2(ℙ) of all real finite linear combinations of random variables of the type

    H_p_1(ξ_1)· H_p_2(ξ_2)⋯ H_p_k(ξ_k)

for k≥ 1, where the integers p_1,...,p_k ≥ 0 satisfy p_1+⋯+p_k = q, and (ξ_1,...,ξ_k) is a standard real Gaussian vector extracted
from A (in particular, C_0 = ℝ).


Taking into account the orthonormality and completeness of ℍ in L^2(γ), together with a standard monotone class argument (see e.g. <cit.>), it is possible to prove that C_q    C_m in  L^2(ℙ) for every q≠ m, and moreover

    L^2(Ω, σ( A), ℙ) = ⊕_q=0^∞ C_q,

that is, every real-valued functional F of A can be (uniquely) represented as a series, converging in L^2, of the form

    F = ∑_q=0^∞ F[q],

where F[q]:= proj(F   |   C_q) stands for the the projection of F onto C_q (F[0]= proj(F   |   C_0) =  [F]).

















  §.§.§ Random nodal lengths



Consider now the integral representation rapSphere for the nodal length of random spherical harmonics. It can be equivalently written as 

    ℒ_ℓ = ∫_𝕊^2δ_0(T_ℓ(x)) ∇ T_ℓ(x) dx
    =√(ℓ(ℓ+1)/2)∫_𝕊^2δ_0(T_ℓ(x)) ∇T_ℓ(x) dx,

where  ∇ is the normalized gradient, i.e. ∇:= ∇ / √(2/ℓ(ℓ +1)) (see 3.2.1 in <cit.> for details). 

We are going to recall the chaotic expansion e:chaos2 for ℒ_ℓ 

    ℒ_ℓ = ∑_q=0^+∞ℒ_ℓ[2q],

where ℒ_ℓ[2q] denotes the orthogonal projection of ℒ_ℓ onto  C_2q. Note that projections on odd chaoses vanish since the integrand functions in rapSphere2 are both even.

In <cit.> the terms of the series on the r.h.s. of chaos_decomp 
are explicitly given (see also <cit.>). 
As it will be clear later in this section, it suffices to deal with the first three terms corresponding to q=0 (the mean of the random variable) and q=1, 2. See <cit.> for more details. 


Let us introduce now two sequences of real numbers {β_2k}_k=0^+∞ and {α_2n,2m}_n,m=0^+∞ corresponding to the chaotic coefficients of the Dirac mass at 0 and the Euclidean norm respectively: for k=0, 1, 2, …

    β_2k := 1/√(2π) H_2k(0),

while for n,m=0, 1, 2, … 

    α _2n,2m:=√(π/2)(2n)!(2m)!/n!m!1/2^n+mp_n+m(1/4),

where p_N is the swinging factorial coefficient

    p_N(x):=∑_j=0^N(-1)^j(-1)^N(
    [ N; j ]) (2j+1)!/(j!)^2x^j.

The first few terms are

    β _0   =1/√(2π),  β _2=-1/√(
    2π),  β _4=3/√(2π),
    α _00   =√(π/2),  α _02=1/2√(π/2),  α _04=-3/8√(π/2).

The chaotic expansion of the nodal length is 

    ℒ_ℓ =∑_q=0^+∞ℒ_ℓ[2q] =    √(ℓ (ℓ
    +1)/2)∑_q=0^∞∑_u=0^q∑_k=0^uα
    _2k,2u-2kβ _2q-2u/(2k)!(2u-2k)!(2q-2u)!×
       ×∫_𝕊^2H_2q-2u(T_ℓ(x))H_2k(∂_1;xT_ℓ(x))H_2u-2k(∂_2;xT_ℓ(x)) dx,

where we use spherical coordinates (colatitude θ , longitude φ) and for x=(θ _x,φ _x) we are using the notation

    ∂_1;x=. (ℓ(ℓ+1)/2)^-1/2·∂/∂θ|
    _θ =θ _x,  ∂_2;x= (ℓ(ℓ+1)/2)^-1/2·. 1/sinθ∂/∂φ| _θ =θ _x,φ
    =φ _x.




















It is obvious that analogous formulas as chaos_decomp_sphere  hold for the chaotic components of the nodal length ℒ_n of arithmetic random waves. 
We are now in a position to give a sketch of the proof of Theorem <ref> (see <cit.> for details).



§ ON THE PROOF OF THEOREM <REF>


 Substituting first_coeff into chaos_decomp_sphere we have

    ℒ_ℓ[0] = 4π·1/2√(2)√(ℓ(ℓ+1)) = [ℒ_ℓ].

For the second chaotic projection, recalling that H_2(t) = t^2-1 and noting that α_20=α_02, we can write 

    ℒ_ℓ[2] = √(ℓ(ℓ+1)/2) (    β_2 α_00/2!∫_𝕊^2 (T_ℓ(x)^2 - 1) dx 
       + β_0 α_20/2!∫_𝕊^2 ( ⟨∇T_ℓ(x)), ∇T_ℓ(x)⟩ - 2 ) dx  ).

A standard application of Green's identity on manifolds[The author thanks Domenico Marinucci for sharing with her this insight] (see also the proof of <cit.>) yields 

    ℒ_ℓ[2] = √(ℓ(ℓ+1)/2) (    β_2 α_00/2!∫_𝕊^2 (T_ℓ(x)^2 - 1) dx 
       + β_0 α_20/2!∫_𝕊^2 ( ⟨∇T_ℓ(x)), ∇T_ℓ(x)⟩ - 2 ) dx  )
    
    =√(ℓ(ℓ+1)/2) (    β_2 α_00/2!∫_𝕊^2 (T_ℓ(x)^2 - 1) dx 
        +β_0 α_20/2!∫_𝕊^2 ( - 2/ℓ(ℓ+1)T_ℓ(x) Δ T_ℓ(x) - 2 ) dx  )
    
    =√(ℓ(ℓ+1)/2) (    β_2 α_00/2!∫_𝕊^2 (T_ℓ(x)^2 - 1) dx 
    + β_0 α_20/2!∫_𝕊^2 ( 2 T_ℓ(x)^2  - 2 ) dx  ),

where the last equality we used the fact that T_ℓ is an eigenfunction of the Laplacian with eigenvalue -ℓ(ℓ+1). From 2chaos1 we have 

    ℒ_ℓ[2] 
    = √(ℓ(ℓ+1)/2) (   β_2 α_00/2!+  β_0 α_20 )∫_𝕊^2 (T_ℓ(x)^2 - 1) dx = 0,

where we used first_coeff. 
Let us now investigate the fourth chaotic component. To this aim, consider the term 

    ℐ_ℓ:=- 1/4√(ℓ(ℓ+1)/2)1/4!∫_𝕊^2 H_4(T_ℓ(x)) dx

whose mean is zero, and whose variance <cit.> is 

    (ℐ_ℓ) = 1/16ℓ(ℓ+1)/21/4! 2 · 4π· 2π∫_0^π/2 P_ℓ(cosθ)^4 sinθ dθ.
 
A careful analysis of the fourth moment  of Legendre polynomials (by means of Hilb's asymptotic formula Hilb) give, as ℓ→ +∞, 

    (ℐ_ℓ ) ∼1/32logℓ

i.e., the variance of this single term is asymptotically equivalent to the total variance varSphere.
Investigating asymptotic moments of product of powers of Legendre polynomials and their derivatives we prove that, as ℓ→ + ∞, 

    ℒ_ℓ[4]/√((ℒ_ℓ[4]) )   and  ℐ_ℓ/√((ℐ_ℓ))

are fully correlated. 
By chaos_decomp, the orthogonality of Wiener chaoses, opiccolo2, ciaociao  and varSphere we have 

    ℒ_ℓ - [ℒ_ℓ]/(ℒ_ℓ) = ℐ_ℓ/√((ℐ_ℓ)) + o_ℙ(1),

o_ℙ(1) denoting a sequence converging to zero in probability, 
meaning that the fourth order chaotic component or more precisely the term ℐ_ℓ “dominates" the whole series. 
Thus in order to understand the asymptotic behavior of the total nodal length it suffices to study the second order fluctuations of ℐ_ℓ. 
Proposition 3.4 in <cit.> states that, as ℓ→ +∞, 

    ℐ_ℓ/√((ℐ_ℓ))→^d Z,

where Z is a standard Gaussian random variable. This concludes the proof.



Inspired by <cit.>, it should be possible to show directly opiccolo by proving in addition that, as ℓ→ +∞,  

    (∑_q=3^+∞ℒ_ℓ[2q]      ) = O(1).




 §.§ Some insight




  §.§.§ Universality of the mean nodal length


Consider point (i) in Remark <ref>. It is immediate to explain this phenomenon in terms of chaotic components, indeed behind calcoli_media there is the following formula

    [ℒ_ℓ] = ℒ_ℓ [0] = √(ℓ(ℓ+1)/2)·β_0 α_0,0·∫_𝕊^2H_0(T_ℓ(x))_ = 1 dx,

where ℓ(ℓ+1)/2 is the variance of the derivatives of T_ℓ. The same formula, accordingly modified, holds on the torus (and in more general settings – see also <cit.>). Of course, the same result can be obtained by Kac-Rice formula <cit.>. 


  §.§.§ Berry's cancellation phenomenon
 
As briefly explained in (ii) Remark <ref>, Berry's cancellation phenomenon concerns the order of magnitude of the asymptotic variance of the nodal length, which turns out to be smaller than expected. This fact can be expressed in terms of chaotic expansions as the vanishing of the second order chaotic component of the nodal length (see 2chaos2). Indeed otherwise the variance of the nodal length would have been of order ℓ, the “natural" one. 
A careful inspection of 2chaos1 reveals that the steps of the proof of 2chaos2 are independent of the underlying manifold, in particular the same result holds for the toral case (i.e. the second chaotic component of the nodal length for arithmetic random waves vanishes <cit.>). More generally, it is natural to guess that the same cancellation phenomenon appears for the nodal volume on so-called isotropic manifolds (compact two-point homogeneous spaces <cit.>) of any dimension (like the hyperspheres <cit.>), and multidimensional tori (see also <cit.>). 
Moreover, this phenomenon has been observed also for other functionals of nodal sets of Gaussian random eigenfunctions, see <cit.> for a complete discussion. 



  §.§.§ Gaussianity: torus vs. sphere
 In <ref> we gave a sketch of the proof of Theorem <ref>: the second order fluctuations of the spherical nodal length ℒ_ℓ are Gaussian. 
From its proof it turns out that ℒ_ℓ
and the term ℐ_ℓ in Iell have the same asymptotic behavior. As already said, the asymptotic Gaussianity of 

    ∫_𝕊^2 H_4(T_ℓ(x)) dx

and hence of ℐ_ℓ was proved in <cit.>. It is now natural to ask for the asymptotic behavior of the corresponding term on the torus, i.e.

    ∫_𝕋^2 H_4(T_n(x)) dx.

Indeed, also in the toral case the fourth chaotic component dominates the series of the total nodal length (see <cit.>). From Lemma 5.2 in <cit.>  we have 

    ∫_𝕋^2 H_4(T_n(x)) dx = 6/𝒩_n ( 1/√(𝒩_n/2)∑_ξ∈Λ_n^+ (|a_ξ|^2 - 1)  )^2 - 3/𝒩_n^2∑_ξ∈Λ_n |a_ξ|^4,

where Λ_n^+ := {ξ=(ξ_1, ξ_2)∈Λ_n : ξ_2 > 0} if √(n) is not an integer, otherwise Λ_n^+ := {ξ=(ξ_1, ξ_2)∈Λ_n : ξ_2 > 0}∪{ (√(n),0)}. From magic_term3 the standard CLT applied to 
the following sum of i.i.d. random variables 

    1/√(𝒩_n/2)∑_ξ∈Λ_n^+ (|a_ξ|^2 - 1)

entails that  magic_term2 is not asymptotically Gaussian. This discussion gives moreover some insight  on the reasons for the nature of the limiting distribution found in Theorem <ref>.



§ FURTHER RELATED WORK




 §.§ Monochromatic random waves

























Consider now the general setting as in <ref>. 
For a generic Riemannian metric on a smooth compact surface ℳ, the eigenvalues λ_j^2 of the Laplace-Beltrami operator (as in eq) are simple and one cannot define a meaningful random model as for the toral or the spherical case, i.e. by endowing   each eigenspace with a Gaussian measure. 
To overcome this, Zelditch introduced  in <cit.> the so-called monochromatic random waves as general approximate models of random Gaussian Laplace eigenfunctions defined on manifolds not necessarily having spectral multiplicities, see also <cit.> for more details. The monochromatic random wave on (ℳ,g) of parameter λ>0 is  the random field

    ϕ_λ(x) := 1/√(dim(H_λ))∑_λ_j∈ [λ, λ + 1] a_j f_j(x),    x∈ℳ,

where the a_j are i.i.d. standard Gaussian random variables, and

    H_λ := ⊕_λ_j ∈ [λ, λ + 1]Ker(Δ_g + λ_j^2 Id),

Id being the identity operator. The field ϕ_λ is centered Gaussian and its covariance kernel is 

    K_λ(x,y) := (ϕ_λ(x), ϕ_λ(y)) = 1/dim(H_λ)∑_λ_j∈ [λ, λ +1] f_j(x) f_j(y),   x,y∈ℳ.

Now fix a point x∈ℳ, and consider the tangent plane T_xℳ to the manifold at x. The pullback Riemannian random wave (see <cit.>) associated with ϕ_λ is the Gaussian random field on T_xℳ defined as

    ϕ_λ^x(u) := ϕ_λ ( exp_x  ( u/λ ) ),     u∈ T_x ℳ,

where exp_x : T_xℳ→ℳ is the exponential map at x. The random field ϕ_λ^x is  centered Gaussian and from covRiem its covariance kernel is 

    K_λ^x(u,v) = K_λ(exp_x  ( u/λ) , exp_x  ( v/λ )  ),     u,v∈ T_x ℳ.
 

A point x ∈ℳ is  of isotropic scaling if, for every positive function λ↦ r(λ) such that r(λ) = o(λ), as λ→∞, one has that

    sup_u,v ∈𝔹(r(λ))  | ∂^α∂^β[  K_λ^x(u,v) -  2π· J_0( u-v_g_x)]| → 0,  λ→∞,

where α, β∈ℕ^2 are multi-indices labeling partial derivatives with respect to u and v, respectively, ·_g_x is the norm on T_xℳ induced by g, and 𝔹(r(λ)) is the corresponding ball of radius r(λ) containing the origin. The manifold ℳ is of isotropic scaling if every x∈ℳ is a point of isotropic scaling, and the convergence in limit is uniform on x for each α, β∈ℕ^2. 


It is difficult to prove directly that a point x is of isotropic scaling, except on flat tori, but 
sufficient conditions about the geodesics through x are known (see <cit.> and the references therein). The condition
that ℳ is a manifold of isotropic scaling is generic in the space of Riemannian metrics
on any smooth compact manifold, see <cit.> for further details.  Note that one can always choose coordinates around x to have g_x = Id, so that the limiting kernel in limit coincides with cov_berry up to a factor. 



 §.§ Nodal lengths: recent results


Let us now fix a C^1-convex body 𝒟⊂^2 (that is: 𝒟 is a compact convex set with C^1-boundary) such that 0∈𝒟 (i.e., the origin belongs to the interior of 𝒟). 
For every x∈ℳ we define

    𝒵_λ,𝒟^x :=  length( (ϕ_λ^x)^-1(0)∩𝒟).

Let x be a point  of isotropic scaling, and assume that coordinates have been chosen around x in such a way that g_x =  Id. Then, as λ→∞,  

    𝒵_λ, 𝒟^x→^d √(2π)· length(B_1^-1(0) ∩𝒟),

where B_1 is Berry's RWM cov_berry with λ_j = 1. 

Theorem <ref> states that the local behavior of zeros of monochromatic random waves is universal  (see also <cit.> for more details). It is not an easy task to find the distribution of the limiting random variable length(B_1^-1(0) ∩𝒟) in Theorem <ref>, so let us understand at least what happens when the domain 𝒟 grows to ^2. For E>0 let us hence consider

    length(B_1^-1(0)∩√(E)·𝒟) = √(E)·length(B_√(E)^-1(0)∩𝒟) =: √(E)·ℒ_E,

where B_√(E) is Berry's RWM cov_berry with λ_j = √(E). 
 We recall now the main results of <cit.> concerning the behavior of ℒ_E, as E→ +∞. Note that meanlength and e:varlength confirm the results predicted by Berry in <cit.>. 

The expectation of the nodal length ℒ_E is 

    [ℒ_E] = area(𝒟) 1/2√(2)√(E),

 the variance of ℒ_E verifies the asymptotic relation

    (ℒ_E) ∼area(𝒟) 1/512πlog E,    E→∞.

Moreover, as E→∞,

    ℒ_E - [ℒ_E]/√((ℒ_E))⟶^d Z,

where Z∼𝒩(0,1) is a standard Gaussian random variable. 

Note that the mean meanlength has the same form as in the toral or spherical case (see Remark <ref>), and the asymptotic variance e:varlength is of lower order than expected – as predicted by Berry in <cit.>. Thanks to the symmetry of nodal lines on the two dimensional sphere <cit.>, the result on the asymptotic variance for the nodal length on 𝕊^2 varSphere obtained by Wigman in <cit.> is of course consistent to Berry's prediction (e:varlength or <cit.>). 


The proof of Theorem <ref> relies on chaotic expansions as for the toral and spherical case (see <ref>). Here, the second order chaotic component does not vanishes identically but it is possible to prove, by means of Green's identity on manifold, that its contribution is negligible. In view of this, the next chaotic component to study is the fourth one which turns out to dominate the whole series, once proved that the contribution of higher order chaoses is negligible (for details see <cit.>). 

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