![Page 1: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/1.jpg)
On the real zeros of randomtrigonometric polynomials
Jürgen Angst
Université de Rennes 1
Workshop - Numerical methods for algebraic curvesRennes, February 19, 2018
![Page 2: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/2.jpg)
Joint work with
Guillaume Poly, Université de Rennes 1.
Viet-Hung Pham, Vietnamese Institute of Advances Studies,Hanoï.
Federico Dalmao, Universidad de la República de Uruguay,Salto.
![Page 3: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/3.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volume
4 Other universality results
![Page 4: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/4.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volume
4 Other universality results
![Page 5: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/5.jpg)
We consider a polynomial Pd ∈ R[X],
Pd(X) = a0 + a1X + . . .+ adXd,
whose coefficients ak are i.i.d. random variables that are centeredwith unit variance i.e.
E[ak] = 0, E[|ak|2] = 1.
Some natural questions
Can we estimate Nd = #Zd where Zd := {t ∈ R, P (t) = 0}?For a fixed degree d? Asymptotically? In mean? almost surely?Are the real zeros localized?
![Page 6: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/6.jpg)
Theorem (Kac, 1943)
If ak ∼ N (0, 1) for all k′s, then as d tends to infinity
E[Nd] =2
πlog(d) + o (log(d)) .
Theorem (Erdös–Offord, 1956)
If ak ∼ B(±1, 1/2) for all k′s, then as d tends to infinity
Nd =2
πlog(d) + o
(log(d)2/3 log log(d)
)with probability 1− o
(1√
log log(d)
).
![Page 7: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/7.jpg)
Theorem (Ibraghimov–Maslova, 1971)
If the ak are centered with unit variance and in the domain ofattraction of the normal law, then as d tends to infinity
E[Nd] =2
πlog(d) + o (log(d)) .
Such an asymptotics is called universal in the sense that it doesnot depend on the particular law of ak.
![Page 8: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/8.jpg)
Theorem (Nguyen–Nguyen–Vu, 2015)
If the ak are centered with unit variance and admit a finite mo-ment of order (2 + ε), then as d goes to infinity
E[Nd] =2
πlog(d) +O(1).
Tao–Vu (2013) : universality of the correlation functions.
Do–Nguyen–Vu (2016), Kabluchko–Flasche (2018) : universalityof the mean number of real zeros of random polynomials of theform Pd(X) =
∑dk=0 akckX
k.
The real zeros are localized near ±1, in fact for all fixed η > 0
E[#{t ∈ Zd ∩ [±1− η,±1 + η]c] = O(1).
![Page 9: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/9.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volume
4 Other universality results
![Page 10: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/10.jpg)
We will consider here a sum of the type
fn(t) =∑k∈Zd
|k|≤n
ak e2πik·t, t ∈ Td = Rd/Zd,
with the same symmetry constraint ak = a−k, so that fn is real-valued.
The random coefficients ak are supposed to be independent,identically distributed, centered with unit variance
E[ak] = 0, E[|ak|2] = 1.
![Page 11: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/11.jpg)
sdfdfdvg
two realizations of fn in dimension d = 1 and degreen = 10 for Gaussian and Bernoulli coefficients.
![Page 12: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/12.jpg)
sdfdfdvg
two realizations of fn in dimension d = 1 and degreen = 100 for Gaussian and Bernoulli coefficients.
![Page 13: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/13.jpg)
a realization of the interface between {fn > 0} (white) and{fn < 0} (blue) for d = 2, n = 30, Gaussian coefficients
![Page 14: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/14.jpg)
a realization of the interface between {fn > 0} (white) and{fn < 0} (blue) for d = 2, n = 100, Gaussian coefficients
![Page 15: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/15.jpg)
on the unfolded torus in dimension d = 2,degree n = 40, Bernoulli coefficients
![Page 16: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/16.jpg)
d = 2, degree n = 200, Bernoulli coefficients
![Page 17: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/17.jpg)
d = 3, degree n = 20, Bernoulli coefficients
![Page 18: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/18.jpg)
![Page 19: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/19.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volume
4 Other universality results
![Page 20: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/20.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volumeLocal universalityGlobal universality
4 Other universality resultsThe case of dependent coefficientsNon universality of the varianceBeyond the analytic case
![Page 21: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/21.jpg)
{fn = 0} ∩B(x, 1/n)
Local study of Hd−1({fn = 0} ∩B
(x, 1n
))
![Page 22: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/22.jpg)
Theorem (A.–Pham–Poly, 2016)If the coefficients ak are independent, identically distributed,centered with unit variance, then for all x ∈ Td, as n goes toinfinity, the sequence of random variables
nd−1 ×Hd−1({fn = 0} ∩B
(x,
1
n
))converges in distribution to an explicit random variable whoselaw does no depend on x, nor on the law of ak.
![Page 23: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/23.jpg)
Heuristics of the proof :
The zeros of fn(t) identify with the ones of (Xn(t))|t|≤1
Xn(t) :=1
nd/2× fn
(x+
t
n
)
=1
nd/2
∑k∈Zd
|k|≤n
ak e2πik·(x+ t
n).
The processes (Xn(t)) converge in distribution with respectto the C1−topology to an universal Gaussian process.
The nodal volume is a continuous functional with respect tothe C1−topology.
![Page 24: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/24.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volumeLocal universalityGlobal universality
4 Other universality resultsThe case of dependent coefficientsNon universality of the varianceBeyond the analytic case
![Page 25: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/25.jpg)
{fn = 0} ∩K
Global study of Hd−1 ({fn = 0} ∩K)
![Page 26: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/26.jpg)
Theorem (A.–Pham–Poly, 2016)If the random coefficients ak are independent, identically distri-buted, centered with unit variance and if K ⊂ Td is a compactset with a smooth boundary, then there exists a universal explicitconstant Cd such that, as n goes to infinity
limn→+∞
1
n× E
[Hd−1 ({fn = 0} ∩K)
]= Cd ×Hd(K).
![Page 27: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/27.jpg)
Heuristics of the proof :
Divide the set K into microscopic boxes (Kjn)j of size 1/n.
In each box Kjn, we have local universality.
We sum the local contributions...
... and exhibit a uniform upper bound
supn,j
E[|Hd−1
({fn = 0} ∩Kn
j
)|1+α
]< +∞.
![Page 28: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/28.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volume
4 Other universality results
![Page 29: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/29.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volumeLocal universalityGlobal universality
4 Other universality resultsThe case of dependent coefficientsNon universality of the varianceBeyond the analytic case
![Page 30: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/30.jpg)
We consider the trigonometric polynomial
fn(t) :=∑
1≤k≤nak cos(kt) + bk sin(kt), t ∈ R,
where the (ak)k≥1 and (bk)k≥1 are centered Gaussian variablewith correlation ρ : N→ R, i.e.
E[aka`] = E[bkb`] =: ρ(|k − `|),
E[akb`] = 0, ∀k, ` ∈ N∗.
![Page 31: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/31.jpg)
Theorem (A.–Dalmao–Poly, 2017)If the correlation function ρ satisfies some mild hypotheses
limn→+∞
1
n× E
[H0 ({fn = 0} ∩K)
]=
1
π√3×H1(K).
This is the exact same asymptotics as in the i.i.d. case
Theorem (A.–Poly 2015, Flasche, 2016)If the coefficients ak are i.i.d. centered with unit variance
limn→+∞
1
n× E
[H0 ({fn = 0} ∩K)
]=
1
π√3×H1(K).
![Page 32: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/32.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volumeLocal universalityGlobal universality
4 Other universality resultsThe case of dependent coefficientsNon universality of the varianceBeyond the analytic case
![Page 33: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/33.jpg)
The asymptotics of the mean number of zeros is universal, butthe asymptotics of the variance is not !
Theorem (Bally–Caramellino–Poly, 2017)If the coefficients ak are i.i.d. centered with unit variance, andhave a density with respect to the Lebesgue measure
limn→+∞
1
n×var
[H0 ({fn = 0} ∩ [0, π])
]= CN (0,1)+
1
30
(E[a4k]− 3
).
![Page 34: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/34.jpg)
1 Random algebraic polynomials in one variable
2 Random trigonometric polynomials
3 Universality of the nodal volumeLocal universalityGlobal universality
4 Other universality resultsThe case of dependent coefficientsNon universality of the varianceBeyond the analytic case
![Page 35: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/35.jpg)
Let φ be a C0, 1−periodic function, piecewise C1+α Hölder.
We consider the random periodic function
fn(t) =
n∑k=1
akφ(kt).
![Page 36: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/36.jpg)
As before, we look at the zeros of fn in a small ball B(xn, 1/n),i.e we consider
Xn(t) =1√n
n∑k=1
ak φ
(k
(xn +
t
n
)).
xn − 1n xn +
1n
xn
![Page 37: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/37.jpg)
Theorem (A.–Poly, 2017)If the coefficients ak are independent, identically distributed,centered with unit variance, with a finite fourth moment, and ifα := limn→+∞ xn ∈ R\Q is diophantine, then the sequence
H0
({fn = 0} ∩B
(xn,
1
n
))converge in distribution to a random variable whose law doesnot depend on α, nor the particular law of ak.
Remark : if α ∈ Q, convergence holds but the limit is no moreuniversal.
![Page 38: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/38.jpg)
la fin
![Page 39: On the real zeros of random trigonometric polynomials · 2018. 3. 4. · The real zeros are localized near 1, in fact for all fixed >0 E[#ft2Z d\[ 1 ; 1+ ]c] = O(1): 1 Random algebraic](https://reader033.vdocuments.us/reader033/viewer/2022051322/6034b1aba969b45bcc30523e/html5/thumbnails/39.jpg)
ANR research project on universality for random nodal domains
http://unirandom.univ-rennes1.fr