Zeros and critical points of monochromatic random waves 06-18-2018 - PowerPoint PPT Presentation

Zeros and critical points of monochromatic random waves 06-18-2018 Yaiza Canzani

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states | ξ | 2 hamiltonian ∆ g g ( x )

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states | ξ | 2 hamiltonian ∆ g g ( x ) h t √ i ∆ g time evolution geodesic flow e

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states | ξ | 2 hamiltonian ∆ g g ( x ) h t √ i ∆ g time evolution geodesic flow e steady states closed geodesics eigenfunctions ψ λ j

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states | ξ | 2 hamiltonian ∆ g g ( x ) h t √ i ∆ g time evolution geodesic flow e steady states closed geodesics eigenfunctions ψ λ j

The setting: ( M n , g ) compact Riemannian manifold, ∂ M = ∅ Classical Quantum T ∗ M L 2 ( M ) states | ξ | 2 hamiltonian ∆ g g ( x ) h t √ i ∆ g time evolution geodesic flow e steady states closed geodesics eigenfunctions ψ λ j

Questions • # { critical points of Ψ λ } λ n • measure( Z Ψ λ ) λ � � • # components of Z Ψ λ λ n • topologies & nestings in Z Ψ λ

Questions • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • measure( Z Ψ λ ) λ � � • # components of Z Ψ λ λ n • topologies & nestings in Z Ψ λ

Questions • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p − → B n λ � � • # components of Z Ψ λ λ n • topologies & nestings in Z Ψ λ

Questions • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p − → B n • T 2 Rudnick-Wigman ’07 λ � � • # components of Z Ψ λ λ n • topologies & nestings in Z Ψ λ

Questions • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p − → B n • T 2 Rudnick-Wigman ’07 λ � � • S n , T n • # components of Z Ψ λ Nazarov-Sodin ’07 ,’16 E − → C n λ n • topologies & nestings in Z Ψ λ

Questions • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p − → B n • T 2 Rudnick-Wigman ’07 λ � � • S n , T n • # components of Z Ψ λ Nazarov-Sodin ’07 ,’16 E − → C n λ n • S n , T n Sarnak-Wigman ’17 • topologies & nestings in Z Ψ λ C-Sarnak ’17

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ 1 � Cov Ψ λ ( x , y ) = E (Ψ λ ( x )Ψ λ ( y )) = ψ λ j ( x ) ψ λ j ( y ) # { λ j = λ } λ j = λ

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ 1 � Cov Ψ λ ( x , y ) = E (Ψ λ ( x )Ψ λ ( y )) = ψ λ j ( x ) ψ λ j ( y ) # { λ j = λ } λ j = λ Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ 1 � Cov Ψ λ ( x , y ) = E (Ψ λ ( x )Ψ λ ( y )) = ψ λ j ( x ) ψ λ j ( y ) # { λ j = λ } λ j = λ Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Lemma Let x 0 ∈ S n or T n . Then, λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology.

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ 1 � Cov Ψ λ ( x , y ) = E (Ψ λ ( x )Ψ λ ( y )) = ψ λ j ( x ) ψ λ j ( y ) # { λ j = λ } λ j = λ Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Lemma Let x 0 ∈ S n or T n . Then, λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology. Ψ ∞ : R n → R is a Gaussian field with 1 � e i � u − v , w � d σ S n − 1 ( w ) Cov Ψ ∞ ( u , v ) = (2 π ) n S n − 1

Random waves: 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j = λ } )1 / 2 λ j = λ 1 � Cov Ψ λ ( x , y ) = E (Ψ λ ( x )Ψ λ ( y )) = ψ λ j ( x ) ψ λ j ( y ) # { λ j = λ } λ j = λ Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Lemma Let x 0 ∈ S n or T n . Then, λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology. Ψ ∞ : R n → R is a Gaussian field with 1 � e i � u − v , w � d σ S n − 1 ( w ) Cov Ψ ∞ ( u , v ) = (2 π ) n S n − 1 Heuristics: (∆ R n + lot ) Ψ x 0 λ = Ψ x 0 and ∆ R n Ψ ∞ = Ψ ∞ . λ

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1)

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1) 1 � Cov Ψ λ ( x , y ) = ψ λ j ( x ) ψ λ j ( y ) # { λ j ∈ [ λ, λ + 1) } λ j ∈ [ λ,λ +1)

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1) 1 � Cov Ψ λ ( x , y ) = ψ λ j ( x ) ψ λ j ( y ) # { λ j ∈ [ λ, λ + 1) } λ j ∈ [ λ,λ +1) Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1) 1 � Cov Ψ λ ( x , y ) = ψ λ j ( x ) ψ λ j ( y ) # { λ j ∈ [ λ, λ + 1) } λ j ∈ [ λ,λ +1) Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Theorem (C-Hanin ’15, ’16) Let x 0 ∈ M. If measure { geodesic loops closing at x 0 } = 0 , then λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology.

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1) 1 � Cov Ψ λ ( x , y ) = ψ λ j ( x ) ψ λ j ( y ) # { λ j ∈ [ λ, λ + 1) } λ j ∈ [ λ,λ +1) Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Theorem (C-Hanin ’15, ’16) Let x 0 ∈ M. If measure { geodesic loops closing at x 0 } = 0 , then λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology. i.e, we get d Ψ x 0 λ ( u ) − → Ψ ∞ ( u ) .

Universality. 1 � Ψ λ = a j ∼ N (0 , 1) a j ψ λ j iid (# { λ j ∈ [ λ,λ +1) } )1 / 2 λ j ∈ [ λ,λ +1) 1 � Cov Ψ λ ( x , y ) = ψ λ j ( x ) ψ λ j ( y ) # { λ j ∈ [ λ, λ + 1) } λ j ∈ [ λ,λ +1) Let Ψ x 0 x 0 + u � � λ ( u ) := Ψ λ . λ Theorem (C-Hanin ’15, ’16) Let x 0 ∈ M. If measure { geodesic loops closing at x 0 } = 0 , then λ →∞ Cov Ψ x 0 lim λ ( u , v ) = Cov Ψ ∞ ( u , v ) , uniformly in u , v ∈ B (0 , R ) in the C ∞ -topology. i.e, we get d Ψ x 0 λ ( u ) − → Ψ ∞ ( u ) . Random wave conjecture: ψ x 0 λ ( u ) has same statistics as Ψ ∞ ( u ) .

Prior results and today’s talk • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p • T 2 − → B n Rudnick-Wigman ’07 λ • S n , T n � � • # components of Z Ψ λ Nazarov-Sodin ’07 ,’16 E − → C n λ n • S n , T n Sarnak-Wigman ’17 • topologies & nestings in Z Ψ λ C-Sarnak ’17

Prior results and today’s talk • S 2 Nicolaescu ’10 • # { critical points of Ψ λ } Cammarota-Marinucci-Wigman ’14 p − → A n λ n Cammarota-Wigman ’15 • S 2 Neuheisel ’00, Wigman ’09, ’10 • measure( Z Ψ λ ) p • T 2 − → B n Rudnick-Wigman ’07 λ • S n , T n � � • # components of Z Ψ λ Nazarov-Sodin ’07 ,’16 E − → C n λ n • ( M , g ) ∗ Sarnak-Wigman ’17 + C-Hanin ’15’16 • topologies & nestings in Z Ψ λ C-Sarnak ’17 + C-Hanin ’15’16