Nodal Sets of High-Energy Arithmetic Random Waves

Transcription

1 Nodal Sets of High-Energy Arithmetic Random Waves Maurizia Rossi UR Mathématiques, Université du Luxembourg Probabilistic Methods in Spectral Geometry and PDE Montréal August 22-26, 2016 M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

2 This talk is mainly based on Non-Universality of nodal length distribution for arithmetic random waves, joint work with Domenico Marinucci, Giovanni Peccati, Igor Wigman and Phase singularities in complex arithmetic random waves, joint work with Federico Dalmao, Ivan Nourdin, Giovanni Peccati. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

3 outline 1 Berry s Random Wave Model 2 Arithmetic Random Waves 3 Nodal Length Mean & Variance Asymptotic Distribution: Non-Universal NCLT 4 Nodal Intersections Number Phase Singularities Mean, Variance and Distribution M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

4 deterministic eigenfunctions (M, g) compact Riemannian manifold of dimension 2 g Laplace-Beltrami operator g f + Ef = 0 eigenvalues E 0 E 1 E 2 E 3... eigenfunctions f 0, f 1, f 2, f 3,... o.b. of L 2 (M) Behavior of f j for large j : the zeroes f 1 j (0) f j real zeroes are disjoint union of smooth curves (length) M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

5 berry s random wave model For generic chaotic surfaces, compare f j W j W j = (W j (x)) x R 2 centered Gaussian field on the plane Cov (W j (x), W j (y)) = J 0 ( E j x y ), x, y R 2. stationary and isotropic! Q : Compare??? Example: nodal length on T = R 2 /Z 2 (the 2-torus). Take a representative planar domain U R 2 and study nodal length of W j restricted to U (r.v.) Expected nodal length per unit area of W j : E j /2 2 Predicted variance : log E j [Berry, 2002] M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

6 toral eigenfunctions T := R 2 /Z 2 2-torus Laplacian f + Ef = 0 Eigenvalues E n = 4π 2 n, n = sum of two integer squares S = {n = λ λ 2 2, λ 1, λ 2 Z} Set of frequencies Λ n = {λ = (λ 1, λ 2 ) Z 2 : λ λ 2 2 = n} Λ n =: N n (grows on average as log n...) Eigenfunctions (λ Λ n ) e λ (x) := e i2π λ,x, x T L 2 -o.b. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

7 Probability measure on S 1 induced by Λ n µ n = 1 N n density-1 sequence {n j } j {n} s.t. δ n λ λ Λ n µ nj dθ/2π. other weak-* partial limit of {µ n } n [Kurlberg-Wigman, 2016]. µ(k) := z k dµ(z), k Z, Fourier coefficients S 1 η [ 1, 1], {n j } j s.t. µ nj (4) η If µ nj dθ/2π, then µ nj (4) 0. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

8 random laplace eigenfunctions on the torus n S T n (x) = 1 Nn λ Λ n a λ e λ (x), x T {a λ } λ Λn iid complex-gaussian except for a λ = a λ ( T n is real!!!) E[a λ ] = 0 E[ a λ 2 ] = 1 T n centered Gaussian random field r n (x y) = Cov (T n (x), T n (y)) = 1 cos(2π λ, x y ), N n λ Λ n x, y T T n stationary ARW approximate Berry s RWM M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

9 nodal length Nodal lines Tn 1 (0) = {x T : T n (x) = 0} disjoint union of smooth curves L n := length(t 1 n (0)) AIM: to study the sequence of random variables {L n } n S M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

10 mean & asymptotic variance Theorem [Rudnick-Wigman, 2008] For fixed n S, E[L n ] = 1 2 En = π2 n. 2 Consistent with the expected nodal length for the RWM! Theorem [Krishnapur-Kurlberg-Wigman, 2013] As N n + E n Var(L n ) = c n (1 + o(1)) Nn 2 c n = 1 + µ n(4) µ n (4) = 4-th Fourier coefficient of µ n M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

11 more about the asymptotic variance µ n (4) 1 variance order of magnitude E n /N 2 n Natural guess: E n /N n Non-Universality Berry s cancellation phenomenon η [ 1, 1], {n j } j s.t. µ nj (4) η Var(L nj ) 1 + η2 512 E nj N 2 n j M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

12 nodal length in wiener chaoses Integral form L n = length(t 1 n (0)) = T δ 0 (T n (x)) T n (x) dx gradient T n = ( 1 T n, 2 T n ), i := / x i (i = 1, 2). L n L 2 (P) and functional of a Gaussian field Wiener-Itô chaos expansion in L 2 (P) L 2 (P) = + q=0 C q L n = + q=0 L n [q] = + q=0 Proj (L n C q ) if q m, then C q C m Cov (L n [q], L n [m]) = 0 0 order projection is the mean: L n [0] = E[L n ]. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

13 chaotic decomposition Proposition L n = E[L n ] + q 2, L n [2q] = T + q=1 L n [q], E[L n ] = 1 2 En. 2 odd chaoses L n [2q + 1] = 0, q 0 En 2 q u u=0 k=0 α 2k,2u 2k β 2q 2u (2k)!(2u 2k)!(2q 2u)! H 2q 2u (T n (x))h 2k ( 1 T n (x))h 2u 2k ( 2 T n (x)) dx, Var( i T n (x)) = E n 2 2 i := i (i = 1, 2). E n M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

14 2nd chaos component vanishes (berry s cancellation) + α 2,0β 0 2! L n [2] = T En 2 ( α0,0 β 2 2! H 2 ( 1 T n (x)) dx + α 0,2β 0 2! T H 2 (T n (x)) dx+ T ) H 2 ( 2 T n (x)) dx Lemma [R. (2015), Marinucci, Peccati, R., Wigman (2016)] Proof L n [2] = 0 H 2 (t) = t 2 1 ( i T n (x)) 2 dx = T n (x) ii T n (x) dx ( ii := 2 / x 2 i ) T T 11 T n + 22 T n = T n = E n T n α 2,0 = α 0,2 M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

15 4th chaos component (variance) L n E[L n ] = L n [4] + q 3 L n [2q] Var(L n ) = Var(L n [4]) + q 3 Var(L n [2q]). Lemma as N n +, Var(L n [4]) lim n + Var(L n ) = 1. L n E[L n ] Var(Ln ) = L n [4] Var(Ln [4]) + o P(1) 4th chaos dominates the whole series. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

16 4th chaos component (distribution) W (n) = L n [4] = W 1 (n) W 2 (n) W 3 (n) W 4 (n) Proposition As N n +, L n [4] = E n 512 N 2 n En 2 ( α0,0 β 4 4! T = 1 n N n /2 ) H 4 (T n (x)) dx +... λ=(λ 1,λ 2 ) Λ n λ 2 >0 ( aλ 2 1 ) n λ 2 1 λ 2 2 λ 1 λ 2 ( ) 1+W 1 (n) 2 2W 2 (n) 2 2W 3 (n) 2 4W 4 (n) 2 +o P (1). M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

17 Lemma For {n j } S s.t. N nj + and µ nj (4) η, W (n j ) d Z(η) = Z 1 Z 2 Z 3 Z 4, where Z(η) is a centered Gaussian vector with covariance Σ = Σ(η) = η η η η η 8. The eigenvalues of Σ are 0, 3 2, 1 η 8, 1+η 4 Σ is singular. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

18 η [0, 1] M η := η 2 (2 (1 + η)x2 1 (1 η)x 2 2), X = (X 1, X 2 ) standard bivariate Gaussian. η 1 η 2 the distributions of M η1 and M η2 are. Theorem [Marinucci, Peccati, R., Wigman (2016)] For {n j } S s.t. N nj + and µ nj (4) η L nj E[L nj ] Var(Lnj ) law M η Non-Universal NonCLT. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

19 nodal intersections as phase singularities n S. T n and T n two independent Gaussian Laplace eigenfunctions. I n := Tn 1 (0) T 1 n (0) I n is the number of zeroes of complex arithmetic random waves! Θ n (x) := 1 Nn λ Λ n v λ e λ (x), x T, {v λ } λ Λn i.i.d. complex-gaussian: R(v λ ) and I(v λ ) i.i.d. N (0, 1) T n (x) = R(Θ n (x)) Tn (x) = I(Θ n (x)) The set of zeroes of Θ n coincides with Tn 1 (0) Phase singularities T 1 n (0) M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

20 complex berry s random wave model Complex ARW approximate complex Berry s RWM complex centered Gaussian field, whose real and imaginary parts are independent (real) RWM Cov (W j (x), W j (y)) = J 0 ( E j x y ), x, y R 2. [Berry, 2002] Expected number of phase singularities E j /(4π) Predicted variance E j log E j M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

21 nodal intersections number Theorem [Dalmao, Nourdin, Peccati, R. (2016)] (i) n S, E[I n ] = E n 4π. Ĩ nj (ii) As N n +, En 2 Var(I n ) = d n (1 + o(1)), Nn 2 d n = 3 µ n(4) π 2 (iii) As N nj + and µ nj (4) η, ( η η 2 2 A + 1 η ) B 2(C 2), 2 A, B, C ind, A law = 2X 2 + 2Y 2 4Z 2 law = B, C law = X 2 + Y 2 (X, Y, Z) N (0, I 3 3 ). M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

22 steps of the proof T n := (T n, T n ) 2-dim. Gaussian field on T 1. I n = δ 0 (T n (x)) J Tn (x) dx (integral form) T 2. I n = E[I n ] + I n [2q] (chaotic expansion) q 1 3. I n [2] = 0 (Berry s cancellation) En 2 4. As N n +, Var(I n [4]) = d n (1 + o(1)) Nn ( ) 2 5. As N n +, Var I n [2q] = o(var(i n [4])) q 3 6. As N n +, Ĩ n = Ĩn[4] + o P (1) 7. NonCentral and NonUniversal asymptotic distribution of Ĩn[4]. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

23 Proposition ( ) Var I n [2q] = o(var(i n [4])), as N n + q 3 }{{} =:proj(i n C 6 ) Sketch of the proof T = disjoint union of little squares Q (translation of Q 0 = [0, 1/M) [0, 1/M), where M E n, along k/m, k Z 2 ) proj(i n C 6 ) = ) proj (I n Q C 6 Q Var(proj(I n C 6 )) = Q,Q Cov ( ) ( )) proj (I n Q C 6, proj I C n Q 6 Split the sum into good and bad pairs of squares M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

24 bad = singular pairs of squares Fix ε 1 > 0 small number Definition [ Rudnick, Wigman (2014)] (Q, Q ) is singular if (x, y) Q Q s.t. r n (x y) > ε 1 or 1 r n (x y) > ε 1 n or 2 r n (x y) > ε 1 n or 11 r n (x y) > ε 1 n or 12 r n (x y) > ε 1 n or 22 r n (x y) > ε 1 n. Q,Q sing. Q,Q sing. ( ) ( Cov proj (I n Q C 6, proj I C n Q 6)) Var ( )) ( ( )) proj (I n Q C 6 Var proj I C n Q 6 meas(b Q ) ( )) E n Var proj (I n Q0 C 6. 1/E n B Q = union of all squares Q s.t. (Q, Q ) is a singular pair. M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

25 Lemma Lemma ( ) meas(b Q ) = O r n (x) 6 dx. T ( )) Var proj (I n Q0 C 6 = O(1). Proof ( )) Var proj (I n Q0 C 6 E[In 2 Q0 ] = = E[In 2 Q0 ] E[I n Q0 ] + E[I n Q0 ]. }{{}}{{} =:A =:B (fine!) A = 2nd fact moment = K 2 (x, y) dxdy = 1 K Q 0 Q 0 M 2 2 (x) dx, Q 0 [ ] K 2 (x) := p (Tn(x),Tn(0))(0, 0)E J Tn (x) J Tn (0) T n (x) = T n (0) = 0, where p (Tn(x),Tn(0)) is the density of (T n (x), T n (0)). M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

26 kac-rice around the origin K 2 (x) det Ω n(x) 1 r n (x) 2, x T, where det(ω n (x)) = E ( E 2 2 ( 1r n (x)) 2 + ( 2 r n (x)) 2 ) =: Ψ 1 r n (x) 2 n (x) Taylor: as x 0 Recall M E n 1 E n Ψ n (x) = E 3 n x 2 + E 4 n O( x 4 ) K 2 (x) E 2 n K 2 (x) dx 1 E 2 Q 0 En 2 n = O(1). M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

27 end of the proof So far ( ) Var I n [2q] q 3 }{{} =:proj(i n C 6 ) ( ) = O En 2 r n (x) 6 dx T Now r n (x) 6 dx = S 6(n) T Nn 6 [Bombieri-Bourgain, 2015] so that q 3 }{{} S 6 (n) = O(N 7/2 n ), ( ) Var I n [2q] = o(en/n 2 n) 2 = o(var(l n [4]). M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

28 Berry (2002) Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature.. J. Phys. A. Bombieri, Bourgain (2015) A problem on sums of two squares. Int. Math. Res. Not. Dalmao, Nourdin, Peccati, Rossi (2016) Phase singularities in complex arithmetic random waves. ArXiv: Krishnapur, Kurlberg, Wigman (2013) Nodal length fluctuations for arithmetic random waves. Ann. of Math. (2) Kurlberg, Wigman (2016) On probability measures arising from lattice points on circles. Math. Ann. Marinucci, Peccati, Rossi, Wigman (2016) Non-Universality of nodal length distribution for arithmetic random waves. Geom. Funct. Anal. Rossi (2015) The geometry of spherical random fields. PhD Thesis. Rudnick, Wigman (2008) On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré. Rudnick, Wigman (2014) Nodal intersections for random eigenfunctions on the torus. Am. J. Math. (to appear) M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29

29 for your attention... M. Rossi (Université du Luxembourg) Random Nodal Sets on the Torus Montréal / 29