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: { f k } k ≥ 1 a sequence of functions acting on some domain Ω, { a k } k ≥ 1 a sequence of i.i.d. random variables with same distribution µ , Z n = { x ∈ Ω | � n k =1 a k f k ( x ) = 0 } . Which aspect of Z n depend on µ as n → ∞ ? � (Expected volume, Euler characteristic, topological properties...) Some functional of Z n 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 k =1 a k X k = 0 with { a k } k ≥ 1 i.i.d. and a 1 ∼ N (0 , 1). � n 1943: Kac E ( N n ( R )) ∼ 2 π log( n ) . 1945, 1956: Erdös, Littlewood, Offord extended Kac’s result to the case P ( a 1 = 1) = P ( a 1 = − 1) = 1 2 . 1971: Ibragimov, Maslova established the first universality result: if E ( a 1 ) = 0 and { a k } k ≥ 1 are in the domain of attraction of the normal Law: E ( N n ( R )) ∼ 2 π log( n ) . � � 1974: Maslova when E ( | a 1 | 2+ ǫ ) and c = 4 1 − 2 : π π Var ( N n ( R )) ∼ c log( n ) , N n ( R ) − E ( N n ( R )) → N (0 , 1) . � c log( n ) 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 other families of polynomials, if E ( a 1 ) = 0 and E ( | a 1 | 2+ ǫ ) < ∞ . �� n z k , E ( N n ( R )) ∼ √ n , � Elliptic: � n k =0 a k k � √ n Weyl: � n k ! z k , E ( N n ( R )) ∼ 2 1 k =0 a k π 2015: F. Dalmao provided variance and CLT for elliptic �� n � x k (Kostlan-Schub-Smale) random polynomials � n k =0 a k k when a k ∼ N (0 , 1), 2017: Y. Do & V. Vu provided variance and CLT for Weyl polynomials when a k ∼ N (0 , 1). Guillaume POLY Universality for roots of random trigonometric models

The case of random trigonometric polynomials n n � � a k cos( kt ) | a k cos( kt ) + b k sin( kt ) k =1 k =1 has been looked for the first time by Dunnage (1966) who proved when a k ∼ N (0 , 1) 2 E ( N n ([0 , 2 π ]) ∼ √ 3 n . 2008: A. Granville & I. Wigman Var ( N n ([0 , 2 π ]) ∼ cn , N n ([0 , 2 π ]) − E ( N n ([0 , 2 π ])) √ 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 of Wiener-Itô expansions and Nualart-Peccati criterion of central convergence. 2016: H. Flasche proved that if E ( a 1 ) = 0, E ( a 2 1 ) = 1 then 2 √ E ( N n ([0 , 2 π ]) ∼ 3 n . 2017: O. Nguyen and V. Vu established local universality and universality of the expected number of roots of weighted trigonometric polynomials: n � a k c k cos( kt ) + b k d k sin( kt ) . k =1 Guillaume POLY Universality for roots of random trigonometric models

2017, V. Bally, L. Caramellino, G. P. Let { a k , b k } k ≥ 1 be i.i.d. random variables that are centred with variance 1 and infinitely many moments and L a 1 ≥ c 1 [ a , b ] ( x ) dx for c > 0 and a < b : � �� C gauss + 1 � E ( a 4 Var ( N n ([0 , π ])) ∼ n 1 ) − 3 30 Unexpected result since the local statistics are universal. � Guillaume POLY Universality for roots of random trigonometric models

Empirical distribution of N n ([0 ,π ]) − E ( N n ([0 ,π ]) when a k ∼ N (0 , 1) √ n (left picture) and a k ∼ 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 : { a k } k ≥ 1 an independent sequence of r.v. with same law µ , centred with unit variance: � S n � √ n , N (0 , 1) d TV − n →∞ 0 − − → ⇔ ∃ n 0 ≥ 1 , L S n 0 is not singular ⇔ µ ∗ µ ∗ · · · ∗ µ is not singular . � �� � n 0 times Many improvements: convergence C k 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 � S n � � � � � � S p e − x 2 dx � � √ p , N (0 , 1) √ n ∈ A − √ d TV = sup � P � 2 � 2 π � A ∈B ( R ) A � S p � < 1 ⇒ S p Hence, d TV √ p , N (0 , 1) not singular . √ p Reciprocally, assume S p √ p is not singular. We can write L S p = c µ AC + (1 − c ) µ S . µ AC is an absolutely continuous probability measure and µ S a singular one and c ∈ ]0 , 1[. Guillaume POLY Universality for roots of random trigonometric models

Intuition behind local CLT Take f ∈ L 1 ( R ) and positive. f ∗ f = lim M →∞ ↑ f ∗ (min( f , M )) . And f ∗ min( f , M ) ∈ C 0 ( R ) as convolution of L 1 and L ∞ mappings. Then, f ∗ f is lower semi-continuous and we can find α > 0, a < b such that f ∗ f ≥ α 1 [ a , b ] . Applying this to the density of µ AC we get L S 2 p L S p ∗ L S p = = µ AC ∗ µ AC + (2 µ AC ∗ µ S + µ S ∗ µ S ) 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: Law = ǫ U + (1 − ǫ ) V , S 2 p 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 n � S n = a k k =1 n   2 p − 1 2( k +1) p � � = a k   k =0 i =2 kp +1 n 2 p � = ǫ k U k + (1 − ǫ k ) V k . k =0 Guillaume POLY Universality for roots of random trigonometric models

Intuition behind CLT By a conditionning argument with respect to { ǫ k , V k } k ≥ 1 one is left to treat the case of independent and uniformly distributed random variables. Assume { U k } k ≥ 1 are i.i.d. with uniform distribution on [ − 1 , 1]. For any φ ∈ C 1 ∩ L ip ( R ) we get � 1 � 1 φ ′ ( x )(1 − x 2 ) dx = 2 x φ ( x ) dx , − 1 − 1 � S n � 1 � � � � n n 1 � � φ ′ (1 − U 2 √ n √ n k ) = 2 φ ( U k ) U k E E n k =1 k =1 Guillaume POLY Universality for roots of random trigonometric models

Intuition behind CLT By the law of large number � S n � 1 � S n � � n � �� � � � φ ′ φ ′ 1 − U 2 1 − U 2 √ n √ n E ≈ E E k k n k =1 � S n � �� 2 φ ′ ≈ √ n 3 E � � n 1 � √ n ≈ E Ψ( U k ) k =1 with Ψ( x ) = 2 x φ ( x ) . φ ′ is continuous and ψ is C 1 then there is regularization effect. CLT is uniform on C 1 class of functions the previous relations makes the convergence uniform on the unit ball of C 0 hence the total variation distance. Guillaume POLY Universality for roots of random trigonometric models

Using local CLT in Kac-Rice formula for expectation Set { a k , b k } k ≥ 1 an i.i.d. sequence of r.v. centred with unit variance and compactly supported continuous density f ( x ) dx . Set f n ( t ) = � n k =1 a k cos( kt ) + b k sin( kt ) and N n = Card { t ∈ [0 , 2 π ] , | F n ( t ) = 0 } . By Kac-Rice formula we have � 2 π � | y | ρ ( n ) E ( N n ) = t (0 , y ) dydt , 0 R with ρ ( n ) t ( x , y ) the density of ( f n ( t ) , f ′ n ( t )) . 1 In order to use the CLT we write instead: F n ( t ) = √ n f n ( t ) and we get � 2 π n � | y | r ( n ) E ( N n ) = ( y , 0) dydt , t 0 R with r ( n ) ( x , y ) the density of ( F n ( t ) , F ′ n ( t )). t Guillaume POLY Universality for roots of random trigonometric models

Using some local CLT (for some weighted uniform metric) we have √ � � 3 � 2 π e − x 2+3 y 2 � � r ( n ) ( t , x , y ) ∈ [0 , 2 π ] × R 2 (1 + | y | 3 ) � ( x , y ) − � � → 0 , C n := sup 2 t � � √ � � � 2 π n � 2 π n � � 1 2 π e − 3 y 2 3 � � | y | ρ ( n ) 2 dydt � t (0 , y ) dydt − | y | � � � n � 0 R 0 R � � 2 π n � | y | C n ≤ 1 + | y | 3 dydt − − − → n →∞ 0 . n 0 R E ( N n ) 2 √ lim = 3 . n n →∞ 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 �� � � n � � � � � � < ǫ , P c k a k � � � k =1 where { a k } k ≥ 1 are i.i.d. and c k deterministic. Through Edgeworth expansions, local CLT approach works for variance estimates. Guillaume POLY Universality for roots of random trigonometric models