nodal sets of high-energy arithmetic random waves · nodal sets of high-energy arithmetic random...

Nodal Sets of High-Energy

Arithmetic Random Waves

Maurizia RossiUR Mathematiques, Universite du Luxembourg

Probabilistic Methods inSpectral Geometry and PDE

Montreal – August 22-26, 2016

M. Rossi (Universite du Luxembourg) Random Nodal Sets on the Torus Montreal–23.08.2016 1 / 29

This talk is mainly based on

Non-Universality of nodal length distribution for arithmetic randomwaves,joint work withDomenico Marinucci, Giovanni Peccati, Igor Wigman

and

Phase singularities in complex arithmetic random waves,joint work withFederico Dalmao, Ivan Nourdin, Giovanni Peccati.


outline

1 Berry’s Random Wave Model

2 Arithmetic Random Waves

3 Nodal LengthMean & VarianceAsymptotic Distribution: Non-Universal NCLT

4 Nodal Intersections NumberPhase SingularitiesMean, Variance and Distribution


deterministic eigenfunctions

(M, g) compact Riemannian manifold of dimension 2

∆g Laplace-Beltrami operator

∆gf + Ef = 0

eigenvalues E0 ≤ E1 ≤ E2 ≤ E3 ≤ . . .

eigenfunctions f0, f1, f2, f3, . . . o.b. of L2(M)

Behavior of fj for large j : the zeroes f−1j (0)

fj real ⇒ zeroes are disjoint union of smooth curves (length)


berry’s random wave model

For “generic” chaotic surfaces, compare

fj ←→ Wj

Wj = (Wj(x))x∈R2 centered Gaussian field on the plane

Cov (Wj(x),Wj(y)) = J0(√Ej‖x− y‖), x, y ∈ R2.

⇒ stationary and isotropic!

Q : Compare???

Example: nodal length on T = R2/Z2 (the 2-torus).

Take a representative planar domain U ⊂ R2 and study nodal lengthof Wj restricted to U (r.v.)

• Expected nodal length per unit area of Wj :√Ej/2

√2

• Predicted variance : ≈ logEj [Berry, 2002]M. Rossi (Universite du Luxembourg) Random Nodal Sets on the Torus Montreal–23.08.2016 5 / 29

toral eigenfunctions

T := R2/Z2 2-torus ∆ Laplacian

∆f + Ef = 0

Eigenvalues En = 4π2n, n = sum of two integer squares

S = n = λ12 + λ2

2, ∃λ1, λ2 ∈ Z

Set of frequencies Λn = λ = (λ1, λ2) ∈ Z2 : λ12 + λ2

2 = n

|Λn| =: Nn (grows on average as√

log n...)

Eigenfunctions (λ ∈ Λn)

eλ(x) := ei2π〈λ,x〉, x ∈ T

L2-o.b.M. Rossi (Universite du Luxembourg) Random Nodal Sets on the Torus Montreal–23.08.2016 6 / 29

Probability measure on S1 induced by Λn

µn =1

Nn

∑λ∈Λn

δ λ√n

∃ density-1 sequence njj ⊂ n s.t.

µnj ⇒ dθ/2π.

∃ other weak-* partial limit of µnn [Kurlberg-Wigman, 2016].

µ(k) :=

∫S1z−k dµ(z), k ∈ Z, Fourier coefficients

∀η ∈ [−1, 1],∃njj s.t.

µnj(4)→ η

If µnj ⇒ dθ/2π, then µnj(4)→ 0.


random laplace eigenfunctions on the torus

n ∈ STn(x) =

1√Nn

∑λ∈Λn

aλeλ(x), x ∈ T

aλλ∈Λn iid complex-Gaussian except for aλ = a−λ(⇒ Tn is real !!!)

E[aλ] = 0 E[|aλ|2] = 1

Tn centered Gaussian random field

rn(x−y) = Cov (Tn(x), Tn(y)) =1

Nn

∑λ∈Λn

cos(2π〈λ, x−y〉), x, y ∈ T

⇒ Tn stationary

• ARW approximate Berry’s RWM


nodal length

Nodal lines T−1n (0) = x ∈ T : Tn(x) = 0

disjoint union of smooth curves

Ln := length(T−1n (0))

AIM: to study the sequence of random variables Lnn∈SM. Rossi (Universite du Luxembourg) Random Nodal Sets on the Torus Montreal–23.08.2016 9 / 29

mean & asymptotic variance

Theorem [Rudnick-Wigman, 2008] For fixed n ∈ S,

E[Ln] =1

2√

2

√En =

1

2√

2

√4π2n.

Consistent with the expected nodal length for the RWM!————————————————————————Theorem [Krishnapur-Kurlberg-Wigman, 2013] As Nn → +∞

Var(Ln) = cnEnN 2n

(1 + o(1))

cn =1 + µn(4)2

512

µn(4) = 4-th Fourier coefficient of µn


more about the asymptotic variance

|µn(4)| ≤ 1 ⇒ variance order of magnitude En/N 2n

• Natural guess: En/Nn Berry’s cancellation phenomenon

• Non-Universality

∀η ∈ [−1, 1],∃njj s.t.

µnj(4)→ η

⇒ Var(Lnj) ∼1 + η2

512

EnjN 2nj


nodal length in wiener chaoses

Integral form

Ln = length(T−1n (0)) =

∫Tδ0(Tn(x))‖∇Tn(x)‖ dx

gradient ∇Tn = (∂1Tn, ∂2Tn), ∂i := ∂/∂xi (i = 1, 2).

Ln ∈ L2(P) and functional of a Gaussian field

———————————————————————————⇒ Wiener-Ito chaos expansion in L2(P)

L2(P) =+∞⊕q=0

Cq ⇒ Ln =+∞∑q=0

Ln[q] =+∞∑q=0

Proj (Ln | Cq)

if q 6= m, then Cq ⊥ Cm ⇒ Cov (Ln[q],Ln[m]) = 0

0− order projection is the mean: Ln[0] = E[Ln].


chaotic decomposition

Ln = E[Ln] ++∞∑q=1

Ln[q], E[Ln] =1

2√

2

√En.

Proposition

odd chaoses Ln[2q + 1] = 0, q ≥ 0

q ≥ 2, Ln[2q] =

√En2

q∑u=0

u∑k=0

α2k,2u−2kβ2q−2u

(2k)!(2u− 2k)!(2q − 2u)!×

×∫TH2q−2u(Tn(x))H2k(∂1Tn(x))H2u−2k(∂2Tn(x)) dx,

Var(∂iTn(x)) =En2→ ∂i :=

√2

En∂i (i = 1, 2).


2nd chaos component vanishes (berry’s cancellation)

Ln[2] =

√En2

(α0,0β2

2!

∫TH2(Tn(x)) dx+

+α2,0β0

2!

∫TH2(∂1Tn(x)) dx+

α0,2β0

2!

∫TH2(∂2Tn(x)) dx

)Lemma [R. (2015), Marinucci, Peccati, R., Wigman (2016)]

Ln[2] = 0

Proof

H2(t) = t2 − 1∫T(∂iTn(x))2 dx = −

∫TTn(x)∂iiTn(x)dx (∂ii := ∂2/∂x2i )

∂11Tn + ∂22Tn = ∆Tn = −EnTnα2,0 = α0,2


4th chaos component (variance)

Ln − E[Ln] = Ln[4] +∑q≥3

Ln[2q]

⇒ Var(Ln) = Var(Ln[4]) +∑q≥3

Var(Ln[2q]).

Lemma

limn→+∞

Var(Ln[4])

Var(Ln)= 1.

⇒ as Nn → +∞,

Ln − E[Ln]√Var(Ln)

=Ln[4]√

Var(Ln[4])+ oP(1)

4th chaos dominates the whole series.M. Rossi (Universite du Luxembourg) Random Nodal Sets on the Torus Montreal–23.08.2016 15 / 29

4th chaos component (distribution)

Ln[4] =

√En2

(α0,0β4

4!

∫TH4(Tn(x)) dx+ ...

)

W (n) =

W1(n)W2(n)W3(n)W4(n)

=1

n√Nn/2

∑λ=(λ1,λ2)∈Λn

λ2>0

(|aλ|2 − 1

)nλ2

1

λ22

λ1λ2

.

PropositionAs Nn → +∞,

Ln[4] =

√En

512N 2n

(1+W1(n)

2−2W2(n)2−2W3(n)

2−4W4(n)2+oP(1)

).


Lemma

For nj ⊆ S s.t. Nnj → +∞ and µnj(4)→ η,

W (nj)d−→ Z(η) =

Z1

Z2

Z3

Z4

,

where Z(η) is a centered Gaussian vector with covariance

Σ = Σ(η) =

1 1

212

012

3+η8

1−η8

012

1−η8

3+η8

00 0 0 1−η

8

.

The eigenvalues of Σ are 0, 32, 1−η

8, 1+η

4⇒ Σ is singular.


η ∈ [0, 1]

Mη :=1

2√

1 + η2(2− (1 + η)X2

1 − (1− η)X22 ),

X = (X1, X2) standard bivariate Gaussian.η1 6= η2 ⇒ the distributions of Mη1 and Mη2 are 6=.

Theorem [Marinucci, Peccati, R., Wigman (2016)]

For nj ⊆ S s.t. Nnj → +∞ and |µnj(4)| → η

Lnj − E[Lnj ]√Var(Lnj)

law→Mη

Non-Universal NonCLT.


nodal intersections as phase singularities

n ∈ S. Tn and Tn two independent Gaussian Laplace eigenfunctions.

In := |T−1n (0) ∩ T−1

n (0)|

• In is the number of zeroes of complex arithmetic random waves!

Θn(x) :=1√Nn

∑λ∈Λn

vλeλ(x), x ∈ T,

vλλ∈Λn i.i.d. complex-Gaussian: <(vλ) and =(vλ) i.i.d. N (0, 1)

→ Tn(x) = <(Θn(x)) Tn(x) = =(Θn(x))

The set of zeroes of Θn coincides with T−1n (0) ∩ T−1

n (0)

Phase singularities


complex berry’s random wave model

Complex ARW approximate complex Berry’s RWMcomplex centered Gaussian field, whose real and imaginary parts areindependent (real) RWM

Cov (Wj(x),Wj(y)) = J0(√Ej‖x− y‖), x, y ∈ R2.

[Berry, 2002]

Expected number of phase singularities Ej/(4π)

Predicted variance ≈ Ej · logEj


nodal intersections number

Theorem [Dalmao, Nourdin, Peccati, R. (2016)]

(i) n ∈ S, E[In] =En4π.

(ii) As Nn → +∞, Var(In) = dnE2n

N 2n

(1 + o(1)),

dn =3µn(4)2 + 5

128π2.

(iii) As Nnj → +∞ and |µnj(4)| → η,

Inj ⇒1

2√

10 + 6η2

(1 + η

2A+

1− η2

B − 2(C − 2)

),

A,B,C ind, Alaw= 2X2 + 2Y 2 − 4Z2 law= B, C

law= X2 + Y 2

(X, Y, Z) ∼ N (0, I3×3).


steps of the proof

Tn := (Tn, Tn) 2-dim. Gaussian field on T

1. In =

∫Tδ0(Tn(x)) |JTn(x)| dx (integral form)

2. In = E[In] +∑q≥1

In[2q] (chaotic expansion)

3. In[2] = 0 (Berry’s cancellation)

4. As Nn → +∞, Var(In[4]) = dnE2n

N 2n

(1 + o(1))

5. As Nn → +∞, Var

(∑q≥3

In[2q]

)= o(Var(In[4]))

6. As Nn → +∞, In = In[4] + oP(1)

7. NonCentral and NonUniversal asymptotic distribution of In[4].


kac-rice around the origin

K2(x) det Ωn(x)

1− rn(x)2, x ∈ T,

where

det(Ωn(x)) =E

2

(E2− (∂1rn(x))2 + (∂2rn(x))2

1− rn(x)2

)=: Ψn(x)

Taylor: as ‖x‖ → 0

Ψn(x) = E3n‖x‖2 + E4

nO(‖x‖4)

⇒ K2(x) E2n

Recall M ≈√En

1

En

∫Q0

K2(x) dx 1

E2n

E2n = O(1).


end of the proof

So far

Var( ∑

q≥3


)= O

(E2n

∫Trn(x)6 dx

)

Now ∫Trn(x)6 dx =

|S6(n)|N 6n

[Bombieri-Bourgain, 2015]

|S6(n)| = O(N 7/2n ),

so that

Var( ∑

q≥3


)= o(E2

n/N 2n) = o(Var(Ln[4]).


Berry (2002) Statistics of nodal lines and points in chaotic quantumbilliards: perimeter corrections, fluctuations, curvature.. J. Phys. A.

Bombieri, Bourgain (2015) A problem on sums of two squares. Int.Math. Res. Not.

Dalmao, Nourdin, Peccati, Rossi (2016) Phase singularities in complexarithmetic random waves. ArXiv: 1608.05631

Krishnapur, Kurlberg, Wigman (2013) Nodal length fluctuations forarithmetic random waves. Ann. of Math. (2)

Kurlberg, Wigman (2016) On probability measures arising from latticepoints on circles. Math. Ann.

Marinucci, Peccati, Rossi, Wigman (2016) Non-Universality of nodallength distribution for arithmetic random waves. Geom. Funct. Anal.

Rossi (2015) The geometry of spherical random fields. PhD Thesis.

Rudnick, Wigman (2008) On the volume of nodal sets foreigenfunctions of the Laplacian on the torus. Ann. Henri Poincare.

Rudnick, Wigman (2014) Nodal intersections for random eigenfunctionson the torus. Am. J. Math. (to appear)


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nodal sets of high-energy arithmetic random waves · nodal sets of high-energy arithmetic random...

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