[PPT] - Asymptotic Distribution of Nodal Intersections for Arithmetic PowerPoint Presentation

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Asymptotic Distribution of Nodal Intersections for Arithmetic Random Waves

Maurizia Rossi

Universit´ e Paris Descartes

Random Waves in Oxford June 18-22, 2018

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This talk is based on a joint work with Igor Wigman King’s College London

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utline

1

Nodal Intersections: Deterministic Results

2

Arithmetic Random Waves

3

Nodal Intersections: Mean and Variance

4

Nodal Intersections: Asymptotic Distribution Chaotic expansions Limit Theorems

5

Some Details: Approximate Kac-Rice Formula

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toral eigenfunctions Standard flat torus T := R2/Z2 Helmholtz equation ∆f + Ef = 0, E > 0 Eigenvalues En = 4π2n, n = λ12 + λ22 ∃ λ1, λ2 ∈ Z Set of frequencies Λn = {λ = (λ1, λ2) ∈ Z2 : λ2

1 + λ2 2 = n}

#Λn =: Nn cardinality of the set of frequencies O.b. of eigenfunctions λ ∈ Λn eλ(x) := ei2πλ,x, x ∈ T f(x) := 1 √Nn

λ∈Λn

aλeλ(x), x ∈ T {aλ}λ∈Λn ⊂ C such that aλ = a−λ.

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nodal intersections / deterministic setting Smooth curve C ⊂ T Nodal intersections f −1(0) ∩ C Theorem [Bourgain-Rudnick, 2012] Let C ⊂ T be real analytic with nowhere zero curvature, then √n 1−o(1) ≪ #f −1(0) ∩ C ≪ √n. Conjecture [Bourgain-Rudnick, 2012] If C ⊂ T is smooth with nowhere zero curvature, then #f −1(0) ∩ C ≫ √n. Theorem [Bourgain-Rudnick, 2015] Let C ⊂ T be smooth with nowhere zero curvature, then for “most” n #f −1(0) ∩ C ≫ √n.

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arithmetic random waves [Oravecz-Rudnick-Wigman, 2008] n sum of two squares Tn(x) := 1 √Nn

λ∈Λn

aλeλ(x), x ∈ T {aλ}λ∈Λn i.d. complex-Gaussian, independent save for aλ = a−λ E[aλ] = 0 E[|aλ|2] = 1 ⇒ Tn centered Gaussian r.f. whose cov. kernel is Cov (Tn(x), Tn(y)) = 1 Nn

λ∈Λn

cos(2πλ, x−y) =: rn(x−y), x, y ∈ T ⇒ Tn is stationary.

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nodal intersections / random setting Nodal line T −1

n (0) = {x ∈ T : Tn(x) = 0}

a.s. smooth curve Smooth curve with nowhere zero curvature C ⊂ T Nodal intersections number Zn := #T −1

n (0) ∩ C

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mean and variance Theorem [Rudnick-Wigman, 2016] Curve C ⊂ T: smooth, with nowhere zero curvature and length L. E[Zn] = √En π √ 2 L. For {n} s.t. Nn → +∞ Var(Zn) = (4BC(Λn) − L2) n Nn + O

n

N 3/2

n

,

where BC(Λn) := L L 1 Nn

λ∈Λn

λ |λ|, ˙ γ(t1) 2 λ |λ|, ˙ γ(t2) 2 dt1dt2 and γ : [0, L]− →C is a unit speed parameterization.

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Probability measure on S1 induced by Λn µn = 1 Nn

λ∈Λn

δ λ

√n

invariant w.r.t. rotations by π/2.

∃ density-1 sequence {nj}j ⊂ {n} s.t.

µnj ⇒ dθ/2π.

[Cilleruelo, 1993] There exists a thin sequence {nj}j ⊂ {n} s.t.

µnj ⇒ 1 4 (δ±1 + δ±i) .

∃ other weak-* partial limits of {µn}n, partially classified by

[Kurlberg-Wigman, 2016].

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n the variance

4BC(Λn) − L2 = 4

S1

L θ, ˙ γ(t)2 dt 2 dµn(θ) − L2. (i) 0 ≤ 4BC(Λn) − L2 ≤ L2; (ii) No (unique) limit as n → +∞; (iii) 4BC(Λn) − L2 can vanish: (1) 4BC(µ) − L2 = 0, ∀ prob. measure µ on S1 invariant w.r.t. rotations by π/2 (universally); (2) 4BC

Λnj
− L2 → 0, for some {nj}j ⊂ {n}.
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integral representation γ : [0, L]− →C unit speed parameterization fn(t) := Tn(γ(t)), t ∈ [0, L] Nodal intersections number Zn = number of zeroes of fn in [0, L] (1) Zn = L δ0(fn(t))|f ′

n(t)| dt,

where f ′

n(t) = ∇Tn(γ(t)), ˙

γ(t). fn(t) and f ′

n(t) are independent for fixed t

Var(f ′

n(t)) = En/2

⇒

f ′

n(t) := f ′ n(t)/

En/2

We rewrite (1) as Zn =

En

2 L δ0(fn(t))| f ′

n(t)| dt.

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nodal intersections and wiener chaos

Zn is an explicit L2(P)-functional of the Gaussian process fn L2(P) =

+∞

q=0

Cq Cq, q = 0, 1, 2, . . . are Wiener chaoses. For q = q′, Cq ⊥ Cq′. (Normalized Hermite polynomials {Hk}k≥0 form an orthonormal basis for the space of L2-functions on R w.r.t. the Gaussian density.)

Wiener-Itˆ
chaos expansion

(i) Zn =

+∞

q=0

Zn[q] = E[Zn] +

+∞

q=1

Zn[q] 1) the series in (i) converges in L2(P); 2) Zn[q] = proj(Zn|Cq) orthogonal proj. onto Cq; 3) if q = q′, then Zn[q] and Zn[q′] are uncorrelated.

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chaotic expansions

Lemma Wiener-Itˆ

expansion

Zn =

En

2 L δ0(fn(t))| f′

n(t)| dt

=

+∞

q=0
En

2

q

ℓ=0

b2q−2ℓa2ℓ L H2q−2ℓ(fn(t))H2ℓ( f′

n(t)) dt

=Zn[2q]

, where Hk, k = 0, 1, · · · ≡ Hermite polynomials and b2q−2ℓ = 1 (2q − 2ℓ)! √ 2πH2q−2ℓ(0) (formal chaotic coefficients of δ0(·)) whereas a2ℓ =

2

π (−1)ℓ+1 2ℓℓ!(2ℓ − 1) (chaotic coefficients of | · |)

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2nd chaotic projections Zn[0] = √En π √ 2 L = E[Zn] Proposition n sum of two squares Zn[2] = Za

n[2] + Zb n[2],

where Za

n[2] :=

√ 2π2n 2π 1 Nn 2

λ∈Λ+

n

(|aλ|2 − 1)

2

L λ |λ|, ˙ γ(t) 2 dt − L

.

We have Var(Za

n[2]) = n

Nn (4BC(Λn) − L2) and, as n → +∞ s.t. Nn → +∞, Var(Zb

n[2]) = o (Var(Za n[2])) .

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clt for the nodal intersections number Theorem [R.-Wigman, 2017] Curve C ⊂ T: smooth, with nowhere zero curvature and length L. For {n} s.t. Nn → +∞ and 4BC(Λn) − L2 bdd. away from 0, Z ∼ N(0, 1) Zn − E[Zn]

Var(Zn)

L

→ Z. Sketch of the proof. As n → +∞, Var(Zn) ∼ Var(Za

n[2])

⇒ Zn − E[Zn]

Var(Zn)

= Za

n[2]

Var(Za

n[2])

+ oP(1). By Lindeberg’s criterion (see previous slide) Za

n[2]

Var(Za

n[2]) L

→ Z.

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further developments What about the case 4BC(Λn) − L2 not bdd away from 0? Static curves 4BC(µ) − L2 = 0, ∀µ prob. meas. on S1 invariant w.r.t. rotations by π/2 Example C the full circle For static curves: (i) No exact asymptotic variance for Zn (upper bound); (ii) Za

n[2] = 0

⇒ Zn[2] = Zb

n[2]

⇒ investigation of Zb

n[2] and the series +∞ q=2 Zn[2q]

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δ-separated sequences of energy levels Def. A sequence {n} of energy levels is δ-separated if min

λ=λ′,λ,λ′∈Λn |λ − λ′| ≫ n1/4+δ.

Lemma [Bourgain-Rudnick, 2011] Fix ε > 0. For all but O(N 1−ε/3) integers n ≤ N min

λ=λ′,λ,λ′∈Λn |λ − λ′| ≫ (√n)1−ε.

[Granville-Wigman, 2017]

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asymptotic variance for static curves Proposition [R.-Wigman, 2017] Let C ⊂ T be a static curve on the torus with nowhere zero curvature, of total length L. Let {n} be a δ-separated sequence such that Nn → +∞, then Var(Zn) = n 4N 2

n

16AC(Λn) − L2

(1 + o(1)), where AC(Λn) = 1 N 2

n

λ,λ′∈Λn

L λ |λ|, ˙ γ(t) 2 λ′ |λ′|, ˙ γ(t) 2 dt 2 . Moreover, 16AC(Λn) − L2 is bounded away from zero.

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some results for δ-separeted sequences In order to find the asymptotic variance in the case of static curves for δ-separeted sequences of energy levels, we need to bound the contribution of the off-diagonal terms coming from the 2nd, the 4th and the 6th moments of the covariance kernel. Lemma [R.-Wigman, 2017] For δ-separeted sequences of energy levels we have, as Nn → +∞,

λ=λ′

1 |λ − λ′| = o(1), 1 N 2

n

λ1+λ2+λ3+λ4=0

1 |λ1 + λ2 + λ3 + λ4| = o(1), 1 N 4

n

λ1+λ2+λ3+λ4+λ5+λ6=0

1 |λ1 + λ2 + λ3 + λ4 + λ5 + λ6| = o(1).

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2nd chaotic component for static curves Lemma For n sum of two squares, if C is static Zn[2] = Za

n[2] + Zb n[2]

Var(Za

n[2]) = 4BC(Λn) − L2 = 0

For δ-separated sequences of energy levels, as Nn → +∞, Var(Zb

n[2]) ≪ n

N 2

n

λ=λ′

1 |λ − λ′|

=o(1)

= o n N 2

n

.
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4th chaotic component for static curves

Proposition For δ-separated sequences {n} and static curve C Zn[4] = Za

n[4] + Zb n[4];

Za

n[4] =

√ 2n 4Nn

−

L

W t

2(n) − 1

L L W u

2 (n) du

2 dt + 4 Nn

λ∈Λn

L λ |λ|, ˙ γ(t) 4 dt − L

,

where W t

2(n) := 1

√

Nn/2

λ∈Λ+

n (|aλ|2 − 1)2 λ

|λ|, ˙

γ(t)2 and, as n → ∞, Var(Za

n[4]) =

n 4N 2

n

(16AC(Λn) − L2) ∼ Var(Zn), Var(Zb

n[4]) = o(Var(Za n[4])

→ 4th chaotic component dominates!

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non-gaussianity for static curves Theorem [R.-Wigman, 2017] Let C ⊂ T be a static curve on the torus with nowhere zero curvature, of total length L. Let {n} be a δ-separated sequence such that Nn → +∞, then Var(Zn) ∼ n 4N 2

n

16AC(Λn) − L2

. If µn ⇒ µ, then

Zn − E[Zn]

Var(Zn)

d

→ 1

16AC(µ) − L2
a(C, µ)(Z2

1 − 1) + b(C, µ)(Z2 2 − 1) + c(C, µ)Z1Z2

,

where Z1, Z2 are i.d.d. standard Gaussian random variables.

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example Let C ⊂ T be a full circle of total length L. Let {n} be a δ-separated sequence such that Nn → +∞. Then Var(Zn) ∼ L2 32 n N 2

n

. Moreover, if µn ⇒ dθ/2π, then

Zn − E[Zn]

Var(Zn)

d

→ 1 − Z2

1 + Z2 2

2 , where Z1, Z2 are i.d.d. standard Gaussian random variables.

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work in progress with maffucci Investigation of nodal intersections on the 3-dimensional torus T3 := R3/Z3 against a surface Σ

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kac-rice formula Zeros of the Gaussian process fn in [0, L] If I ⊆ [0, L] s.t. (fn(t1), fn(t2)) is non-degenerate for every t1 = t2 ∈ I, then Var(Zn(I)) =

I×I

(K2(t1, t2) − K1(t1)K1(t2)) dt1dt2 + E[Zn]

K1(t) := φfn(t)(0)E[|f′

n(t)||fn(t) = 0]

K2(t1, t2) := φ(fn(t1),fn(t2))(0, 0)E[|f′

n(t1)| · |f′ n(t2)||fn(t1) = fn(t2) = 0],

for t1 = t2 I1, I2 ⊂ [0, L] disjoint, s.t. (fn(t1), fn(t2)) is non-degenerate ∀t1 ∈ I1, t2 ∈ I2 Cov (Zn(I1), Cov (Zn(I2)) =

I1×I2

(K2(t1, t2) − K1(t1)K1(t2)) dt1dt2.

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Problem (fn(t1), fn(t2)) is degenerate also for some t1 = t2, t1, t2 ∈ [0, L]. Idea (inspired by [Rudnick-Wigman, 2016]) [0, L] =

i

Ii ⇒ Zn =

i

Zn(Ii) Ii × Ij singular if rn(t1 − t2) = ±1, for some t1 ∈ Ii, t2 ∈ Ij Var(Zn) =

i,j

Cov (Zn(Ii), Zn(Ij)) =

i,j singular

Cov (Zn(Ii), Zn(Ij)) +

i,j non-singular

Cov (Zn(Ii), Zn(Ij))

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non-singular vs singular For the non-singular part we can apply Kac-Rice formula

i,j non-singular

Cov (Zn(Ii), Cov (Zn(Ij)) =

i,j non-singular
Ii×Ij

(K2(t1, t2) − K1(t1)K1(t2)) dt1dt2. Then we perform a Taylor expansion of the integrand. For the singular part

i,j singular

Cov (Zn(Ii), Zn(Ij))

≪ n · meas
i,j singular

Ii × Ij

≪ n ·

L rn(t)6 dt.

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approximate kac-rice formula for static curves Proposition [R.-Wigman, 2017] Let C ⊂ T be a static smooth curve on the torus with nowhere zero curvature, of total length L. Let {n} be a δ-separated sequence such that Nn → +∞, then

Var(Zn) = n L L 3 4r4 + 1 12(r12/α)4 − (r2/√α)4 4 − (r1/√α)4 4 + 2(r12/α)r(r1/√α)(r2/√α) + (r1/√α)2(r2/√α)2 2 − 3 2r2(r2/√α)2 − 3 2r2(r1/√α)2 + 1 2(r12/α)2r2 + 1 2(r2/√α)2(r12/α)2 + 1 2(r1/√α)2(r12/α)2 dt1dt2 + o n N 2

n

.
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some references

Bourgain and Rudnick (2011) On the nodal sets of toral eigenfunctions. Inventiones Math. Krishnapur, Kurlberg and Wigman (2013) Nodal length fluctuations for arithmetic random waves. Ann. of Math. (2). Kurlberg and Wigman (2016) On asymptotic angular distributions of lattice points lying in the circle. Math. Ann. Marinucci, Peccati, R. and Wigman (2016) Non-universality on nodal length distribution for arithmetic random waves. Geom. Func. Anal.

R. and Wigman (2017) Asymptotic distribution of nodal intersections for

arithmetic random waves. ArXiv:1702.05179. Rudnick and Wigman (2016) Nodal intersections for random eigenfunctions on the torus. Amer. J. Math.

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thank you for your attention!

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