Asymptotic Distribution of Nodal Intersections for Arithmetic - PowerPoint PPT Presentation

Asymptotic Distribution of Nodal Intersections for Arithmetic Random Waves Maurizia Rossi Universit´ e Paris Descartes Random Waves in Oxford June 18-22, 2018 M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 1 / 30

This talk is based on a joint work with Igor Wigman King’s College London M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 2 / 30

outline Nodal Intersections: Deterministic Results 1 Arithmetic Random Waves 2 Nodal Intersections: Mean and Variance 3 Nodal Intersections: Asymptotic Distribution 4 Chaotic expansions Limit Theorems Some Details: Approximate Kac-Rice Formula 5 M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 3 / 30

toral eigenfunctions T := R 2 / Z 2 Standard flat torus Helmholtz equation ∆ f + Ef = 0 , E > 0 n = λ 12 + λ 22 E n = 4 π 2 n, Eigenvalues ∃ λ 1 , λ 2 ∈ Z Λ n = { λ = ( λ 1 , λ 2 ) ∈ Z 2 : λ 2 1 + λ 2 Set of frequencies 2 = n } #Λ n =: N n cardinality of the set of frequencies e λ ( x ) := e i 2 π � λ,x � , O.b. of eigenfunctions λ ∈ Λ n x ∈ T � 1 f ( x ) := √N n a λ e λ ( x ) , x ∈ T λ ∈ Λ n { a λ } λ ∈ Λ n ⊂ C such that a λ = a − λ . M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 4 / 30

nodal intersections / deterministic setting Nodal intersections f − 1 (0) ∩ C Smooth curve C ⊂ T 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. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 5 / 30

arithmetic random waves [Oravecz-Rudnick-Wigman, 2008] n sum of two squares � 1 √N n x ∈ T T n ( x ) := a λ e λ ( x ) , λ ∈ Λ n { a λ } λ ∈ Λ n i.d. complex-Gaussian, independent save for a λ = a − λ E [ | a λ | 2 ] = 1 E [ a λ ] = 0 ⇒ T n centered Gaussian r.f. whose cov. kernel is � Cov ( T n ( x ) , T n ( y )) = 1 cos(2 π � λ, x − y � ) =: r n ( x − y ) , x, y ∈ T N n λ ∈ Λ n ⇒ T n is stationary. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 6 / 30

nodal intersections / random setting T − 1 Nodal line n (0) = { x ∈ T : T n ( x ) = 0 } a.s. smooth curve Smooth curve with nowhere zero curvature C ⊂ T Z n := # T − 1 n (0) ∩ C Nodal intersections number M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 7 / 30

mean and variance Theorem [Rudnick-Wigman, 2016] Curve C ⊂ T : smooth, with nowhere zero curvature and length L . √ E n E [ Z n ] = √ 2 L. π For { n } s.t. N n → + ∞ � � Var( Z n ) = (4 B C (Λ n ) − L 2 ) n n + O , N n N 3 / 2 n where � L � L � λ � 2 � λ � 2 � 1 B C (Λ n ) := | λ | , ˙ γ ( t 1 ) | λ | , ˙ γ ( t 2 ) dt 1 dt 2 N n 0 0 λ ∈ Λ n and γ : [0 , L ] − →C is a unit speed parameterization. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 8 / 30

Probability measure on S 1 induced by Λ n � µ n = 1 δ λ N n √ n λ ∈ Λ n invariant w.r.t. rotations by π/ 2 . • ∃ density-1 sequence { n j } j ⊂ { n } s.t. µ n j ⇒ dθ/ 2 π. • [Cilleruelo, 1993] There exists a thin sequence { n j } j ⊂ { n } s.t. µ n j ⇒ 1 4 ( δ ± 1 + δ ± i ) . • ∃ other weak-* partial limits of { µ n } n , partially classified by [Kurlberg-Wigman, 2016]. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 9 / 30

on the variance �� L � 2 � 4 B C (Λ n ) − L 2 = 4 γ ( t ) � 2 dt dµ n ( θ ) − L 2 . � θ, ˙ S 1 0 (i) 0 ≤ 4 B C (Λ n ) − L 2 ≤ L 2 ; (ii) No (unique) limit as n → + ∞ ; (iii) 4 B C (Λ n ) − L 2 can vanish: (1) 4 B C ( µ ) − L 2 = 0 , ∀ prob. measure µ on S 1 invariant w.r.t. rotations by π/ 2 (universally); � � − L 2 → 0 , for some { n j } j ⊂ { n } . (2) 4 B C Λ n j M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 10 / 30

integral representation γ : [0 , L ] − →C unit speed parameterization f n ( t ) := T n ( γ ( t )) , t ∈ [0 , L ] Nodal intersections number Z n = number of zeroes of f n in [0 , L ] � L δ 0 ( f n ( t )) | f ′ Z n = n ( t ) | dt, (1) 0 where f ′ n ( t ) = �∇ T n ( γ ( t )) , ˙ γ ( t ) � . f n ( t ) and f ′ n ( t ) are independent for fixed t � � Var( f ′ n ( t ) := f ′ n ( t )) = E n / 2 ⇒ f ′ n ( t ) / E n / 2 � � L E n δ 0 ( f n ( t )) | � f ′ We rewrite (1) as Z n = n ( t ) | dt. 2 0 M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 11 / 30

nodal intersections and wiener chaos Z n is an explicit L 2 ( P ) -functional of the Gaussian process f n + ∞ � L 2 ( P ) = C q q =0 C q , q = 0 , 1 , 2 , . . . are Wiener chaoses. For q � = q ′ , C q ⊥ C q ′ . (Normalized Hermite polynomials { H k } k ≥ 0 form an orthonormal basis for the space of L 2 -functions on R w.r.t. the Gaussian density.) • Wiener-Itˆ o chaos expansion + ∞ + ∞ � � ( i ) Z n = Z n [ q ] = E [ Z n ] + Z n [ q ] q =0 q =1 1) the series in ( i ) converges in L 2 ( P ) ; 2) Z n [ q ] = proj ( Z n | C q ) orthogonal proj. onto C q ; 3) if q � = q ′ , then Z n [ q ] and Z n [ q ′ ] are uncorrelated. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 12 / 30

chaotic expansions Lemma Wiener-Itˆ o expansion � � L E n δ 0 ( f n ( t )) | � f ′ Z n = n ( t ) | dt 2 0 � � L q + ∞ � � E n H 2 q − 2 ℓ ( f n ( t )) H 2 ℓ ( � f ′ = b 2 q − 2 ℓ a 2 ℓ n ( t )) dt , 2 0 q =0 ℓ =0 � �� � = Z n [2 q ] where H k , k = 0 , 1 , · · · ≡ Hermite polynomials and 1 √ (formal chaotic coefficients of δ 0 ( · ) ) b 2 q − 2 ℓ = 2 πH 2 q − 2 ℓ (0) (2 q − 2 ℓ )! whereas � ( − 1) ℓ +1 2 a 2 ℓ = (chaotic coefficients of | · | ) 2 ℓ ℓ !(2 ℓ − 1) π M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 13 / 30

2 nd chaotic projections √ E n √ Z n [0] = 2 L = E [ Z n ] π Proposition n sum of two squares Z n [2] = Z a n [2] + Z b n [2] , where � � √ � L � λ � 2 � 2 π 2 n 1 ( | a λ | 2 − 1) Z a dt − L n [2] := 2 2 | λ | , ˙ γ ( t ) . N n 2 π 0 λ ∈ Λ + n We have n [2]) = n Var( Z a (4 B C (Λ n ) − L 2 ) N n and, as n → + ∞ s.t. N n → + ∞ , Var( Z b n [2]) = o (Var( Z a n [2])) . M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 14 / 30

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. N n → + ∞ and 4 B C (Λ n ) − L 2 bdd. away from 0 , Z ∼ N (0 , 1) Z n − E [ Z n ] L � → Z. Var( Z n ) Sketch of the proof. As n → + ∞ , Var( Z n ) ∼ Var( Z a n [2]) Z a ⇒ Z n − E [ Z n ] n [2] � = � + o P (1) . Var( Z n ) Var( Z a n [2]) By Lindeberg’s criterion (see previous slide) Z a n [2] L � → Z. Var( Z a n [2]) M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 15 / 30

further developments What about the case 4 B C (Λ n ) − L 2 not bdd away from 0 ? Static curves 4 B C ( µ ) − L 2 = 0 , ∀ µ prob. meas. on S 1 invariant w.r.t. rotations by π/ 2 Example C the full circle For static curves: (i) No exact asymptotic variance for Z n (upper bound); (ii) Z a Z n [2] = Z b n [2] = 0 ⇒ n [2] n [2] and the series � + ∞ ⇒ investigation of Z b q =2 Z n [2 q ] M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 16 / 30

δ -separated sequences of energy levels Def. A sequence { n } of energy levels is δ -separated if λ � = λ ′ ,λ,λ ′ ∈ Λ n | λ − λ ′ | ≫ n 1 / 4+ δ . min Lemma [Bourgain-Rudnick, 2011] Fix ε > 0 . For all but O ( N 1 − ε/ 3 ) integers n ≤ N λ � = λ ′ ,λ,λ ′ ∈ Λ n | λ − λ ′ | ≫ ( √ n ) 1 − ε . min [Granville-Wigman, 2017] M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 17 / 30

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 N n → + ∞ , then � 16 A C (Λ n ) − L 2 � n Var( Z n ) = (1 + o (1)) , 4 N 2 n where �� L � 2 � λ � 2 � λ ′ � 2 � 1 A C (Λ n ) = | λ | , ˙ γ ( t ) | λ ′ | , ˙ γ ( t ) dt . N 2 n 0 λ,λ ′ ∈ Λ n Moreover, 16 A C (Λ n ) − L 2 is bounded away from zero. M. Rossi ( Paris 5 ) Nodal Intersections on the Torus Oxford – June 21, 2018 18 / 30