Second Order Results for Nodal Sets of Gaussian Random Waves Giovanni Peccati (Luxembourg University) Joint works with: F. Dalmao, G. Dierickx, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman Random Waves in Oxford — June 20, 2018 1 / 1
I NTRODUCTION ⋆ In recent years, proofs of second order results in the high- energy limit (like central and non-central limit theorems) for local quantities associated with random waves on surfaces , like the flat 2-torus, the sphere or the plane (but not only!). Works by J. Benatar, V. Cammarota, F. Dalmao, D. Marinucci, I. Nourdin, G. Peccati, M. Rossi, I. Wigman. ⋆ Common feature: the asymptotic behaviour of such local quantities is dominated (in L 2 ) by their projection on a fixed Wiener chaos , from which the nature of the fluctuations is inherited. ⋆ ‘Structural explanation’ of cancellation phenomena first de- tected by Berry (plane, 2002) and Wigman (sphere, 2010). 2 / 1
V IGNETTE : W IENER C HAOS ⋆ Consider a generic separable Gaussian field G = { G ( u ) : u ∈ U } . ⋆ For every q = 0, 1, 2..., set � � � � : d ◦ p ≤ q P q : = v . s . G ( u 1 ) , ..., G ( u r ) p . Then: P q ⊂ P q + 1 . ⋆ Define the family of orthogonal spaces { C q : q ≥ 0 } as C 0 = R and C q : = P q ∩ P ⊥ q − 1 ; one has ∞ � L 2 ( σ ( G )) = C q . q = 0 ⋆ C q = q th Wiener chaos of G . 3 / 1
A R IGID A SYMPTOTIC S TRUCTURE For fixed q ≥ 2, let { F k : k ≥ 1 } ⊂ C q (with unit variance). ⋆ Nourdin and Poly (2013) : If F k ⇒ Z , then Z has necessarily a density (and the set of possible laws for Z does not depend on G ). ⋆ Nualart and Peccati (2005) : F k ⇒ Z ∼ N ( 0, 1 ) if and only if E F 4 k → 3 (= E Z 4 ) . ⋆ Peccati and Tudor (2005) : Componentwise convergence to Gaussian implies joint convergence. ⋆ Nourdin, Nualart and Peccati (2015) : given { H k } ⊂ C p , then F k , H k are asymptotically independent if and only if Cov ( H 2 k , F 2 k ) → 0. ⋆ Nonetheless, there exists no full characterisation of the asymp- totic structure of chaoses ≥ 3. 4 / 1
B ERRY ’ S R ANDOM W AVES (B ERRY , 1977) ⋆ Fix E > 0. The Berry random wave model on R 2 , with parameter E , written B E = { B E ( x ) : x ∈ R 2 } , is the unique (in law) centred, isotropic Gaussian field on R 2 such that ∆ B E + E · B E = 0, where ∆ = ∂ 2 + ∂ 2 . ∂ x 2 ∂ x 2 1 2 ⋆ Equivalently, � √ √ E � x − y , z � dz = J 0 ( S 1 e i E [ B E ( x ) B E ( y )] = E � x − y � ) . (this is an infinite-dimensional Gaussian object). ⋆ Think of B E as a “canonical” Gaussian Laplace eigenfunc- tion on R 2 , emerging as a universal local scaling limit for arithmetic and monochromatic RWs, random spherical har- monics ... . 5 / 1
N ODAL S ETS Focus on the length L E of the nodal set : B − 1 E ( { 0 } ) ∩ Q : = { x ∈ Q : B E ( x ) = 0 } , where Q is some fixed domain , as E → ∞ . Images: D. Belyaev 6 / 1
A C ANCELLATION P HENOMENON ⋆ Berry (2002): an application of Kac-Rice formulae leads to � E E [ L E ] = area Q × 8 , and a legitimate guess for the order of the variance is √ Var ( L E ) ≍ E . ⋆ However, Berry showed that Var ( L E ) ∼ area Q 512 π log E , whereas the length variances of non-zero level sets display the √ “correct" order of E . ⋆ Such a variance reduction “... results from a cancellation whose meaning is still obscure... ” (Berry (2002), p. 3032). 7 / 1
S PHERICAL C ASE ⋆ Berry’s constants were confirmed by I. Wigman (2010) in the related model of random spherical harmonics — see Domenico’s talk. ⋆ Here, the Laplace eigenvalues are the integers n ( n + 1 ) , n ∈ N . Picture: A. Barnett 8 / 1
A RITHMETIC R ANDOM W AVES (O RAVECZ , R UDNICK AND W IGMAN , 2007) ⋆ Let T = R 2 / Z 2 ≃ [ 0, 1 ) 2 be the 2-dimensional flat torus. ⋆ We are again interested in real (random) eigenfunctions of ∆ , that is, solutions of the Helmholtz equation ∆ f + E f = 0, for some adequate E > 0 ( eigenvalue ). ⋆ The eigenvalues of ∆ are therefore given by the set { E n : = 4 π 2 n : n ∈ S } , where S = { n : n = a 2 + b 2 ; a , b ∈ Z } . ⋆ For n ∈ S , the dimension of the corresponding eigenspace is N n = r 2 ( n ) : = # Λ n , where Λ n : = { ( λ 1 , λ 2 ) : λ 2 1 + λ 2 2 = n } 9 / 1
A RITHMETIC R ANDOM W AVES (O RAVECZ , R UDNICK AND W IGMAN , 2007) We define the arithmetic random wave of order n ∈ S as: 1 a λ e 2 i π � λ , x � , x ∈ T , √N n ∑ f n ( x ) = λ ∈ Λ n where the a λ are i.i.d. complex standard Gaussian, except for the relation a λ = a − λ . We are interested in the behaviour, as N n → ∞ , of the total nodal length L n : = length f − 1 n ( { 0 } ) . Picture: J. Angst & G. Poly 10 / 1
N ODAL L ENGTHS AND S PECTRAL M EASURES ⋆ Crucial role played by the set of spectral probability mea- sures on S 1 1 N n ∑ √ n ( dz ) , µ n ( dz ) : = n ∈ S δ λ / λ ∈ Λ n (invariant with respect to z �→ z and z �→ i · z .) ⋆ The set { µ n : n ∈ S } is relatively compact and its adherent points are an infinite strict subset of the class of invariant probabilities on the circle (see Kurlberg and Wigman (2015)). 11 / 1
A NOTHER C ANCELLATION √ E n ⋆ Rudnick and Wigman (2008): For every n ∈ S , E [ L n ] = √ 2 . � � 2 E n / N 1/2 Moreover, Var ( L n ) = O . Conjecture: Var ( L n ) = n O ( E n / N n ) . ⋆ Krishnapur, Kurlberg and Wigman (2013): if { n j } ⊂ S is such that N n j → ∞ , then E n j Var ( L n j ) = × c ( n j ) + O ( E n j R 5 ( n j )) , N 2 n j where � � � µ n j ( 4 ) 2 1 + � T | r n j ( x ) | 5 dx = o 1/ N 2 c ( n j ) = ; R 5 ( n j ) = . n j 512 ⋆ Two phenomena: (i) cancellation , and (ii) non-universality . 12 / 1
N EXT S TEP : S ECOND O RDER R ESULTS ⋆ For E > 0 and n ∈ S , define the normalized quantities L E : = L E − E ( L E ) L n : = L n − E ( L n ) � � and Var ( L n ) 1/2 . Var ( L E ) 1/2 ⋆ Question : Can we explain the above cancellation phenom- ena and, as E , N n → ∞ , establish limit theorems of the type LAW LAW � � L E − → Y , and − → Z ? L n ′ j ( { n ′ j } ⊂ S is some subsequence) 13 / 1
A C OMMON S TRATEGY ⋆ Step 1 . Let V = f n or B E , and L = L E or L n . Use the representation (based on the coarea formula) � in L 2 ( P ) , δ 0 ( V ( x )) �∇ V ( x ) � dx , L = to deduce the Wiener chaos expansion of L . ⋆ Step 2 . Show that exactly one chaotic projection L ( 4 ) : = proj ( L | C 4 ) dominates in the high-energy limit – thus ac- counting for the cancellation phenomenon. ⋆ Step 3 . Study by “bare hands” the limit behaviour of L ( 4 ) . 14 / 1
F LUCTUATIONS FOR B ERRY ’ S M ODEL Theorem (Nourdin, P., & Rossi, 2017) 1. (Cancellation) For every fixed E > 0 , proj ( L E | C 2 q + 1 ) = 0, q ≥ 0, and proj ( � L E | C 2 ) reduces to a “negligible boundary term”, as E → ∞ . 2. ( 4 th chaos dominates) Let E → ∞ . Then, � L E = proj ( � L E | C 4 ) + o P ( 1 ) . 3. (CLT) As E → ∞ , � L E ⇒ Z ∼ N ( 0, 1 ) . 15 / 1
R EFORMULATION ON G ROWING D OMAINS Theorem Define, for B = B 1 : L r : = length ( B − 1 ( { 0 } ) ∩ Ball ( 0, r )) . Then, 1. E [ L r ] = π r 2 √ 2 ; 2 2. as r → ∞ , Var ( L r ) ∼ r 2 log r 256 ; 3. as r → ∞ , L r − E [ L r ] Var ( L r ) 1/2 ⇒ Z ∼ N ( 0, 1 ) . 16 / 1
F LUCTUATIONS FOR A RITHMETIC R ANDOM W AVES Theorem (Marinucci, P., Rossi & Wigman, 2016) 1. (Exact Cancellation) For every fixed n ∈ S, proj ( L n | C 2 ) = proj ( L n | C 2 q + 1 ) = 0, q ≥ 0. 2. ( 4 th chaos dominates) Let { n j } ⊂ S be such that N n j → ∞ . Then, L n j = proj ( � � L n j | C 4 ) + o P ( 1 ) . 3. (Non-Universal/Non-Gaussian) If | � µ n j ( 4 ) | → η ∈ [ 0, 1 ] , µ n ( 4 ) = � z 4 µ n ( dz ) , then where � � � 1 � 2 − ( 1 − η ) Z 2 1 − ( 1 + η ) Z 2 L n j ⇒ M ( η ) : = � , 2 1 + η 2 2 where Z 1 , Z 2 independent standard normal. 17 / 1
P HASE S INGULARITIES Theorem (Dalmao, Nourdin, P. & Rossi, 2016) For � T an independent copy, consider I n : = # [ T − 1 T − 1 n ( { 0 } ) ∩ � n ( { 0 } )] . 1. As N n → ∞ , µ n j ( 4 ) 2 + 5 Var ( I n ) ∼ E 2 3 � n N 2 128 π 2 n 2. If | � µ n j ( 4 ) | → η ∈ [ 0, 1 ] , then � 1 + η � A + 1 − η 1 � I n j ⇒ J ( η ) : = � B − 2 ( C − 2 ) 10 + 6 η 2 2 2 2 with A , B , C independent s.t. A law = B law = 2 X 2 1 + 2 X 2 2 − 4 X 2 3 and C law = X 2 1 + X 2 2 , where ( X 1 , X 2 , X 3 ) is standard Gaussian. 18 / 1
E LEMENTS OF P ROOF (BRW) ⋆ In view of Green’s identity, one has that � 1 proj ( L E | C 2 ) = √ ∂ Q B E ( x ) �∇ B E ( x ) , n ( x ) � dx , 2 E where n ( x ) is the outward unit normal at x (variance bounded). L E | C 4 ) is a l.c. of 4 th order terms, among ⋆ The term proj ( � which � √ V E : = Q H 4 ( B E ( x )) dx , E for which one has that � 24 E Q ) 2 J 0 ( � x − y � ) 4 dxdy ∼ 18 Var ( V E ) = π 2 log E , √ E ( � 2 using e.g. J 0 ( r ) ∼ π r cos ( r − π /4 ) , r → ∞ . ⋆ In the proof, one cannot a priori rely on the “full correlation phenomenon” seen in Domenico’s talk. 19 / 1
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