Universality for roots of random trigonometric models Guillaume - - PowerPoint PPT Presentation

Universality for roots of random trigonometric models

Guillaume POLY 19 june 2018

Guillaume POLY Universality for roots of random trigonometric models

A general question: {fk}k≥1 a sequence of functions acting on some domain Ω, {ak}k≥1 a sequence of i.i.d. random variables with same distribution µ, Zn = {x ∈ Ω | n

k=1 akfk(x) = 0}.

• Which aspect of Zn depend on µ as n → ∞ ?

(Expected volume, Euler characteristic, topological properties...)

• Some functional of Zn which is independent of the underlying

randomness will be called universal.

Guillaume POLY Universality for roots of random trigonometric models

Some history on random polynomials

1938: Littlewood & Offord

n

k=1 akX k = 0 with {ak}k≥1 i.i.d. and a1 ∼ N(0, 1).

1943: Kac E (Nn(R)) ∼ 2 π log(n). 1945, 1956: Erdös, Littlewood, Offord extended Kac’s result to the case P(a1 = 1) = P(a1 = −1) = 1

2.

1971: Ibragimov, Maslova established the first universality result: if E(a1) = 0 and {ak}k≥1 are in the domain of attraction of the normal Law: E (Nn(R)) ∼ 2 π log(n). 1974: Maslova when E(|a1|2+ǫ) and c = 4

π

• 1 − 2

π

• :

Var (Nn(R)) ∼ c log(n), Nn(R) − E (Nn(R))

• c log(n)

→ N(0, 1).

Guillaume POLY Universality for roots of random trigonometric models

2014: Tao & Vu established several universality results at microscopic scales and for the expectation of real roots for

• ther families of polynomials, if E(a1) = 0 and

E(|a1|2+ǫ) < ∞.

Elliptic: n

k=0 ak

n

k

• zk, E (Nn(R)) ∼ √n,

Weyl: n

k=0 ak

• 1

k!zk, E (Nn(R)) ∼ 2 π

√n

2015: F. Dalmao provided variance and CLT for elliptic (Kostlan-Schub-Smale) random polynomials n

k=0 ak

n

k

xk

when ak ∼ N(0, 1), 2017: Y. Do & V. Vu provided variance and CLT for Weyl polynomials when ak ∼ N(0, 1).

Guillaume POLY Universality for roots of random trigonometric models

The case of random trigonometric polynomials

n

• k=1

ak cos(kt) |

n

• k=1

ak cos(kt) + bk sin(kt) has been looked for the first time by Dunnage (1966) who proved when ak ∼ N(0, 1) E (Nn([0, 2π]) ∼ 2 √ 3n. 2008: A. Granville & I. Wigman Var (Nn([0, 2π]) ∼ cn, Nn([0, 2π]) − E (Nn([0, 2π])) √cn → N(0, 1).

Guillaume POLY Universality for roots of random trigonometric models

2014: J.M. Azais, F. Dalmao & J.R. León provided alternate proofs and dealt with the “cosine” case using the framework

• f Wiener-Itô expansions and Nualart-Peccati criterion of

central convergence. 2016: H. Flasche proved that if E(a1) = 0, E(a2

1) = 1 then

E (Nn([0, 2π]) ∼ 2 √ 3n. 2017: O. Nguyen and V. Vu established local universality and universality of the expected number of roots of weighted trigonometric polynomials:

n

• k=1

akck cos(kt) + bkdk sin(kt).

Guillaume POLY Universality for roots of random trigonometric models

2017, V. Bally, L. Caramellino, G. P. Let {ak, bk}k≥1 be i.i.d. random variables that are centred with variance 1 and infinitely many moments and La1 ≥ c1[a,b](x)dx for c > 0 and a < b: Var (Nn([0, π])) ∼ n

• Cgauss + 1

30

• E(a4

1) − 3

• Unexpected result since the local statistics are universal.

Guillaume POLY Universality for roots of random trigonometric models

Empirical distribution of Nn([0,π])−E(Nn([0,π])

√n

when ak ∼ N(0, 1) (left picture) and ak ∼ X 3 with X ∼ N(0, 1) (right picture).

Guillaume POLY Universality for roots of random trigonometric models

Local central limit Theorems

• Central limit Theorem may be strengthened under additional

non degeneracy assumptions on the coefficients Prohorov 1952: {ak}k≥1 an independent sequence of r.v. with same law µ, centred with unit variance: dTV

Sn

√n, N(0, 1)

• −

− − →

n→∞ 0

⇔ ∃n0 ≥ 1, LSn0is not singular ⇔ µ ∗ µ ∗ · · · ∗ µ

• n0 times

is not singular. Many improvements: convergence Ck of densities, convergence in entropy, convergence of Fisher information, relaxation of independence... See Bally, Bobkov, Caramellino, Chistyakov, Götze, Johnson...

Guillaume POLY Universality for roots of random trigonometric models

Intuition behind local CLT

Recall that dTV

• Sp

√p , N(0, 1)

• =

sup

A∈B(R)

• P

Sn

√n ∈ A

• −
• A

e− x2

2

dx √ 2π

• Hence, dTV

Sp

√p, N(0, 1)

• < 1 ⇒ Sp

√p

not singular. Reciprocally, assume Sp

√p is not singular. We can write

LSp = cµAC + (1 − c)µS. µAC is an absolutely continuous probability measure and µS a singular one and c ∈]0, 1[.

Guillaume POLY Universality for roots of random trigonometric models

Intuition behind local CLT

Take f ∈ L1(R) and positive. f ∗ f = lim

M→∞ ↑ f ∗ (min(f , M)) .

And f ∗ min(f , M) ∈ C0(R) as convolution of L1 and L∞

• mappings. Then, f ∗ f is lower semi-continuous and we can

find α > 0, a < b such that f ∗ f ≥ α1[a,b]. Applying this to the density of µAC we get LS2p = LSp ∗ LSp = µAC ∗ µAC + (2µAC ∗ µS + µS ∗ µS) = c 1[a,b] b − a + (1 − c)ν

Guillaume POLY Universality for roots of random trigonometric models

Intuition behind CLT

We have the following decomposition: S2p

Law

= ǫU + (1 − ǫ)V , where U has uniform distribution on [a, b], V has any distribution, ǫ has a Bernoulli distribution with some parameter p ∈]0, 1[ and (ǫ, U, V ) are independent. We may write Sn =

n

• k=1

ak =

n 2p −1

• k=0

 

2(k+1)p

• i=2kp+1

ak

 

=

n 2p

• k=0

ǫkUk + (1 − ǫk)Vk.

Guillaume POLY Universality for roots of random trigonometric models

Intuition behind CLT

By a conditionning argument with respect to {ǫk, Vk}k≥1 one is left to treat the case of independent and uniformly distributed random variables. Assume {Uk}k≥1 are i.i.d. with uniform distribution on [−1, 1]. For any φ ∈ C1 ∩ Lip(R) we get

1

−1

φ′(x)(1 − x2)dx =

1

−1

2xφ(x)dx, E

• φ′

Sn

√n

1

n

n

• k=1

(1 − U2

k)

• =

E

• 1

√n

n

• k=1

2φ(Uk)Uk

• Guillaume POLY

Universality for roots of random trigonometric models

Intuition behind CLT

By the law of large number E

• φ′

Sn

√n

1

n

n

• k=1

1 − U2

k

• ≈

E

• 1 − U2

k

• E
• φ′

Sn

√n

• ≈

2 3E

• φ′

Sn

√n

• ≈

E

• 1

√n

n

• k=1

Ψ(Uk)

• with Ψ(x) = 2xφ(x).

φ′ is continuous and ψ is C1 then there is regularization effect. CLT is uniform on C1 class of functions the previous relations makes the convergence uniform on the unit ball of C0 hence the total variation distance.

Guillaume POLY Universality for roots of random trigonometric models

Using local CLT in Kac-Rice formula for expectation

Set {ak, bk}k≥1 an i.i.d. sequence of r.v. centred with unit variance and compactly supported continuous density f (x)dx. Set fn(t) = n

k=1 ak cos(kt) + bk sin(kt) and

Nn = Card{t ∈ [0, 2π], | Fn(t) = 0}. By Kac-Rice formula we have E (Nn) =

2π

• R

|y|ρ(n)

t (0, y)dydt,

with ρ(n)

t (x, y) the density of (fn(t), f ′ n(t)).

In order to use the CLT we write instead: Fn(t) =

1 √nfn(t)

and we get E (Nn) =

2πn

• R

|y|r (n)

t

(y, 0)dydt, with r (n)

t

(x, y) the density of (Fn(t), F ′

n(t)).

Guillaume POLY Universality for roots of random trigonometric models

Using some local CLT (for some weighted uniform metric) we have Cn := sup

(t,x,y)∈[0,2π]×R2(1 + |y|3)

• r (n)

t

(x, y) − √ 3 2π e− x2+3y2

2

• → 0,

1 n

• 2πn
• R

|y|ρ(n)

t (0, y)dydt −

2πn

• R

|y| √ 3 2π e− 3y2

2 dydt

• ≤

Cn n

2πn

• R

|y| 1 + |y|3 dydt − − − →

n→∞ 0.

lim

n→∞

E (Nn) n = 2 √ 3.

Guillaume POLY Universality for roots of random trigonometric models

Previous method for expectation is robust and applies almost verbatim for a great variety of models: as soon as there is an underlying CLT. Other methods exists for treating the expected number of real zeros:

Flasche, Ibragimov, Kabluchko, Maslova, Zaporozhets... count the number of sign changes Do, Nguyen, Tao, Vu... use complex analysis and Jensen formula

Both methods face anti-concentration problems namely P

• n
• k=1

ckak

• < ǫ
• ,

where {ak}k≥1 are i.i.d. and ck deterministic. Through Edgeworth expansions, local CLT approach works for variance estimates.

Guillaume POLY Universality for roots of random trigonometric models

Step 1: approximate Kac-Rice formula

f ∈ C1([a, b]) with |f | + |f ′| > 0. Then, Card {t ∈ [a, b] | f (t) = 0} := N(f , [a, b]) = lim

δ→0

b

a

|f ′(x)|1{|f (x)|<δ} dx 2δ Set ω = inf[a,b] (|f | + |f ′|), proof reveals that δ < ω ⇒ N([a, b]) = 1 2δ

b

a

|f ′(x)|1{|f (x)|<δ}dx. Estimating inf[0,2π] (|Fn(t)| + |F ′

n(t)|) gives that for δn = 1 n5 :

lim

n→∞

1 nVar(Nn) − 1 nVar

1

2δn

2πn

|F ′

n(t)|1{|Fn(t)|<δn}dt

• = 0.

Guillaume POLY Universality for roots of random trigonometric models

Step2: removing the diagonal

The density of the vector (F ′

n(t), Fn(t), F ′ n(s), Fn(s)) becomes

degenerate when t ≈ s. To bypass this problem, one needs to show: lim

ǫ→0 lim sup n→∞

1 n

• [0,nπ2],|t−s|<ǫ

Cov (Φ(Fn, t), Φ(Fn, s)) dtds = 0, where Φ(Fn, t) =

1 2δn |F ′ n(t)|1{|Fn(t)|<δn}.

The main ingredient is the next estimate of repulsion of zeros: P (Nn([a, a + ǫ]) > p) ≤ C

ǫα

p3 + e−cn

• with α > 1.

Set Ik,ǫ = [kǫ, (k + 1)ǫ], hence the diagonal part becomes: 1 n

⌊ nπ

ǫ ⌋

• k=1

E (Nn(Ik,ǫ)Nn(Ik+1,ǫ)) ≤ C

• ǫα−1 + ne−cn

ǫ

• .

Guillaume POLY Universality for roots of random trigonometric models

Step3: Edgeworth expansion:

standard CLT is not enough

As before, set Φ(Fn, t) =

1 2δn |F ′ n(t)|1{|Fn(t)|<δn}. We have the

following heuristic: 1 n

nπ nπ

|E (Φ(Fn, t)Φ(Fn, s)) − E (Φ(Gn, t)Φ(Gn, s))| dtds ≈ 1 n × n2 × |E (Φ(Fn, ·)Φ(Fn, ·)) − E (Φ(Gn, ·)Φ(Gn, ·))| This requires n × |E (Φ(Fn, ·)Φ(Fn, ·)) − E (Φ(Gn, ·)Φ(Gn, ·))| → 0, whereas standard Berry-Essen bound associated with CLT usually gives: |E (Φ(Fn, ·)Φ(Fn, ·)) − E (Φ(Gn, ·)Φ(Gn, ·))| = O( 1 √n).

Guillaume POLY Universality for roots of random trigonometric models

Edgeworth expansions

Set (Xi)i≥1 an i.i.d. sequence of r.v., centred with unit

• variance. Assume further that X1 ∈ L5(P) and write

µ3 = E(X 3), µ4 = E(X 4), µ5 = E(X 5). Assume that X1 admits a continuous density. Set φ the standard Gaussian density, the Edgeworth expansion in the local CLT at the order 3 gives: P

Sn

√n ≤ x

• −

1 √ 2π

x

−∞

e− t2

2 dt

= φ(x) √n µ3H2(x) 6 + φ(x) n φ(x)

• (µ4 − 3)H3(x)

24 + µ2

3H5(x)

72

• +

φ(x) n

3 2

• µ5

120H4(x) + µ3µ4 144 H6(x) + µ3

3

1296H8(x)

• + Ox

1

n2

• .

Guillaume POLY Universality for roots of random trigonometric models

Edgeworth expansions

Set hn(x) =

1 4δ2

n |x1|1|x2|<δ|x3|1|x4|<δ and ρn(x) the density of

(G′

n(t), Gn(t), G′ n(s), Gn(s)).

E (Φ(Fn, t)Φ(Fn, s)) =

• R4 hn(x)ρn(x)
• 1 + Qn,t,s(x)

√n + Rn,t,s(x) n

• dx

+ Rn,t,s, Qn,t,s(x) and Rn,t,s are explicit polynomials of degree less than 6 whose coefficients involves moments of the coefficients {ak, bk}k≥1 lim

ǫ→0 lim sup n→∞

1 n

• {[0,nπ]2,|t−s|>ǫ}

Rn,t,sdtds = 0.

Guillaume POLY Universality for roots of random trigonometric models

Edgeworth expansions

The rest of computations relies heavily on the next ergodics theorems: lim

n→∞

1 n

n

• k=1

k

n

i

f (kx) = 1 (2i + 1)2π

2π

f (x)dx; x π / ∈ Q Estimates of the following kind are required: ∀x ∈]0, π[,

• 1

n

n

• k=1

ki ni cos(kx)

• ≤ C

nx .

Guillaume POLY Universality for roots of random trigonometric models

Beyond the non-degeneracy assumptions on the coefficients

What happens if the joint distribution of {ak, bk}k≥1 is not smooth? There is no Edgeworth expansion in full generality. Take {ak}k≥1 an i.i.d. with La1 = 1

2 (δ1 + δ−1).

• P

Sn

√n ∈ [−ǫ, ǫ]

• −

ǫ

−ǫ

e− x2

2

dx √ 2π

• ∼ C

√n, whereas Edgeworth expansions suggest a speed of 1

n for

symmetric distributions.

Guillaume POLY Universality for roots of random trigonometric models

Entering the proofs of Edgeworth expansions, one is left to

• btain, for n large enough, estimates of the form:

1 n

n

• k=1

|Φ (ξ cos(kt))| ≤ 1 − Ct |ξ|β , Φ(u) = E

• eiua1

If the coefficients are distributed according to 1

2 (δ1 + δ−1),

1 n

n

• k=1

|cos (ξ cos(kt))| ≤ 1 − Ct |ξ|β . For instance if p is a prime number larger than 5, we have ∀β > 2 p − 3, 1 n

n

• k=1
• cos
• ξ cos(2πk

p )

• ≤ 1 − Cp

|ξ|β . (Subspace Theorem on Diophantine approximation) It should be possible to obtain similar estimates for Lebesgue almost t ∈]0, π[...

Guillaume POLY Universality for roots of random trigonometric models

Some open problems

Does the same estimate hold in full generality for all distributions? Do we have a CLT? What is the behaviour of the variance for r.v. without fourth moment?

Guillaume POLY Universality for roots of random trigonometric models

General periodic signal

{an}n≥1 i.i.d. r.v such that: E (a1) = 0, E

a2

1

= 1,

Fn(t) =

n

• k=1

akf (kt) with f ∈ C0

2π ∩ Lip

Typically one is interested in triangular signal

Guillaume POLY Universality for roots of random trigonometric models

Some local universality result

• J. Angst, G.P. (2018):

Let Fn(t) := 1 √n

n

• k=1

akf

k(pn + t)

n

• ,

with pn

n → θ ∈]0, 1[ diophantine, a1 ∈ L4(P). Let Nn([a, b])

the number of roots of Fn in [a, b] we get Nn([a, b])

law

− − − →

n→∞ Card {t ∈ [a, b] | X∞(t) = 0} ,

with X∞ a stationary Gaussian process with correlation ρ(x) = 1 x

x

f ⋆ ˇ f (t)dt, avec ˇ f (x) = f (−x).

Guillaume POLY Universality for roots of random trigonometric models

Some ideas of the proofs:

We want to establish that Fn(t) converge in distribution in C1 towards X∞. We cannot directly use Kolmogorov criterion E

• F′

n(t) − F′ n(s)

• p ≤ C|t − s|1+ǫ.

We are back to the definition: for λ > 0: lim

δ→0 lim sup n→∞

P

  

sup

(t, s) ∈ [a, b]2 |t − s| ≤ δ

• F′

n(t) − F′ n(s)

• ≥ λ

   = 0.

Guillaume POLY Universality for roots of random trigonometric models

We choose Dn a sufficiently large set in [a, b] verifying Card(Dn) ≤ Cnθ and P

  

sup

(t, s) ∈ [a, b]2 |t − s| ≤ δ

• F′

n(t) − F′ n(s)

• ≥ λ

  

≈ P

  

sup

(t, s) ∈ D2

n

|t − s| ≤ δ

• F′

n(t) − F′ n(s)

• ≥ λ

   .

Guillaume POLY Universality for roots of random trigonometric models

A CLT in exponential dimension Kato,Chernozukhov

We consider X1, · · · , Xn des vecteurs indépendants de Rp de coordonnées notées (Xi,j)1≤j≤p: (A1) For all (i, j) ∈ {1, · · · , n} × {1, · · · , p}: Xi,j ∈ L4(P), (A2) Assume there b > 0 s.t. ∀n ≥ 1, ∀j ∈ {1, · · · , p}, 1 n

n

• i=1

E

• X 2

i,j

• ≥ b,

(A3) Assume for C > 0, ∀n ≥ 1, ∀j ∈ {1, · · · , p}, 1 n

n

• i=1

E

• X 4

i,j

• ≤ C,

(A4) Assume that ∀n ≥ 1, ∀i ∈ {1, · · · , n}, E

• max

1≤j≤p |Xi,j|4

• ≤ 2C.

Guillaume POLY Universality for roots of random trigonometric models

Then denoting Y1, · · · , Yn a Gaussian vector with same covariance as X1, · · · , Xn we have for all (t1, · · · , tp) ∈ Rp:

• P
• SX

n

√n ∈ ] − ∞, t1] × · · · ×] − ∞, tp]

• −

P

• SY

n

√n ∈ ] − ∞, t1] × · · · ×] − ∞, tp]

• ≤ K
• log(pn)

7 6 + log(pn)

n

1 6

• ,

with SX

n

√n = 1 √n

n

• k=1

Xk SY

n

√n = 1 √n

n

• k=1

Yk

Guillaume POLY Universality for roots of random trigonometric models

It suffices to apply the latter Xk = ak

• f ′

k(pn + t)

n

• − f ′

k(pn + s)

n

• (t,s)∈D2

n

. which have polynomial dimension in n We can get ak ∼ N(0, 1) which is easier to deal with.

Guillaume POLY Universality for roots of random trigonometric models