THE LERAY MEASURE OF NODAL SETS FOR RANDOM EIGENFUNCTIONS ON THE TORUS FERENCORAVECZ, ZEE´V RUDNICKAND IGOR WIGMAN Abstract. WestudynodalsetsfortypicaleigenfunctionsoftheLapla- cian on the standard torus in d 2 dimensions. Making use of the ≥ multiplicitiesinthespectrumoftheLaplacian,weputaGaussian mea- 7 sure on the eigenspaces and use it to average over the eigenspace. We 0 consider a sequenceof eigenvalues with growing multiplicity . N →∞ 0 ThequantitythatwestudyistheLeray,ormicrocanonical, measure 2 ofthenodalset. WeshowthattheexpectedvalueoftheLeraymeasure n ofaneigenfunctionisconstant,equalto1/√2π. Ourmainresultisthat a thevariance of Leray measure is asymptotically 1/4π , as , at N N →∞ J least in dimensions d=2 and d 5. ≥ 3 2 v 2 Contents 7 0 1. Introduction 2 9 1.1. Background 2 0 1.2. Leray measure 2 6 0 1.3. Results 3 / 1.4. Related work 3 h p 1.5. About the proof of Theorem 1.1 4 - 1.6. Acknowledgements 4 h 2. Random eigenfunctions on the torus 4 t a 2.1. The basic setup 4 m 2.2. A non-degeneracy condition 5 : v 2.3. Gaussian ensembles 6 i X 2.4. The singular set 7 3. The Leray measure 8 r a 4. The expected value of 10 L 4.1. A formal treatment 10 4.2. A rigorous proof 11 5. A formula for the variance of 12 L 5.1. A formal derivation 13 5.2. Integrability of the kernel 13 5.3. Proof of Theorem 5.1 14 6. The asymptotics of the variance 16 6.1. Singular points 17 6.2. The contribution of Bc 18 6.3. The contribution of the singular set B 19 6.4. A bound for the Hessian of u on a cube 19 6.5. The contribution of a singular cube 20 Date: January 2, 2007. 1 2 FERENCORAVECZ,ZEE´VRUDNICKANDIGORWIGMAN 7. Bounding the fourth moment of the two-point function 21 Appendix A. The intersection of the singular set with codimension one hyperplanes 23 Appendix B. The intersection of the singular set with codimension two hyperplanes 26 References 29 1. Introduction 1.1. Background. Thenodal setofafunctionisthesetofpointswherethe function vanishes. In this paper we study the nodal sets of eigenfunctions of the Laplacian ∆ = d ∂2 on the (standard) flat torus Rd/Zd, d 2. j=1 ∂x2j ≥ Of course we have the simple eigenfunctions such as cos(2π(mx+ny)) or P sin(2πmx)sin(2πny) with corresponding Laplace eigenvalue 4π2(m2 +n2), for which the nodal set have a very simple structure. However, on the standard torus such eigenfunctions are atypical, because the eigenvalues on the torus always have multiplicities. The dimension = (E) of an N N eigenspacecorrespondingtoeigenvalue4π2E isthenumberofintegervectors λ Zd so that λ 2 = E. In dimension d 5 this grows as E roughly ∈ | | ≥ → ∞ as Ed2−1 but has more erratic behaviour for small d, particularly for d = 2. We wish to study the nodal sets of typical eigenfunctions. For this we consider a random eigenfunction on the torus, that is a random linear com- bination 1 f(x)= b cos2πi λ,x c sin2πi λ,x λ λ √2 h i− h i N λ∈ZdX:|λ|2=E with b ,c N(0,1) real Gaussians of zero mean and variance 1 which are λ λ ∼ independent save for the relations b = b , c = c . λ λ λ λ We denote by E( ) the expected v−alue of th−e qua−ntity in this ensemble. • • For instance, the expected amplitude of f is E(f(x)2) = 1. | | 1.2. Leray measure. The fundamental quantity that we study here is the Leray measure, or microcanonical measure, of the nodal set of a function f in our ensemble. This is defined as (see [10, Chapter III], [16, 3.3]) § 1 (1.1) (f):= lim meas x T: f(x) < ǫ . L ǫ 02ǫ { ∈ | | } → and in fact we can define a measure on the nodal set by 1 lim φ(x)dx ǫ→02ǫ Zx:|f(x)|<ǫ which in statistical mechanics is the microcanonical ensemble. This mea- sure also appears in number theory as the “singular integral” in the Hardy- Littlewood method and elsewhere, see e.g. [7, 4]. We may formally write (f)= δ(f(x))dx . L Td Z LERAY MEASURE FOR NODAL SETS OF EIGENFUNCTIONS ON THE TORUS 3 As is well known, the limit (1.1) exists when f = 0 on the nodal set, in ∇ 6 which case dσ(x) (f)= L f(x) Z{x:f(x)=0} |∇ | where dσ is the Riemannian hypersurfacemeasure on the nodalset (see 4). § 1.3. Results. The expected value of (f) turns out to be constant (Theo- L rem 4.1): 1 E( ) = . L √2π To compare, the expected volume (or hypersurface measure) of the nodal set of f in our ensemble is √E for some constant depending only on d d I I the dimension [18]. Our main result concerns the variance of (f) as : L N → ∞ Theorem 1.1. In dimensions d= 2 and d 5, as , ≥ N → ∞ 1 Var( (f)) . L ∼ 4π N We refer to [18] for estimates on the variance of the volume of the nodal sets. Concerningremainderterms,indimensiond = 2weshowthatVar( (f)) = L 1/4π +O(1/ 2). In dimension d 3, we prove Var( (f)) = 1/4π + O(EdN−23+ǫ/ 2)N, for all ǫ > 0. Thus w≥henever > Ed−23L+δ for some δN> 0 N N (which is always valid in dimension d 5), then we get an asymptotic. In ≥ dimensions d = 3,4 we are only able to show that the variance is bounded by O(1/ ), though we believe that the conclusion of Theorem 1.1 holds in N those cases as well. It is somewhat surprising that the result depends only on the dimension of the eigenspace and not on the way the frequencies λ are distributed. In dimension d 5, the directions λ/λ of the frequencies are uniformly ≥ | | distributed on the sphere Sd 1 [17]. However, in two dimensions this need − not be the case (though it holds for most values of E, see [8, 12, 9]). For instance there is an infinite sequence of eigenvalues where the dimension of the eigenspace goes to infinity but the set of directions λ/λ S1 tends to | | ∈ an average of four equally spaced point masses [6]. 1.4. Related work. The study of nodal lines of random waves goes back to Longuet-Higgins [13, 14] who computed various statistics of nodal lines for Gaussian random waves in connection with the analysis of ocean waves. Berry [2] suggested to model highly excited quantum states for classically chaotic systems by using various random wave models, and also computed fluctuations of various quantities in these models (see e.g. [3]). See also Zelditch [20]. The idea of averaging over a single eigenspace in the presence of multiplicities appears in B´erard [1] who computed the expected surface measure of the nodal set for eigenfunctions of the Laplacian on spheres. Neuheisel [15] also worked on the sphere and studied the statistics of Leray measure. He gave an upper bound for the variance, which we believe is not sharp. 4 FERENCORAVECZ,ZEE´VRUDNICKANDIGORWIGMAN 1.5. About the proof of Theorem 1.1. We compute thesecond moment E( 2) by means of Gaussian integration as an integral over the torus L 1 dx E( 2) = L 2π Td 1 u(x)2 Z − where p 1 u(x) := E(f(x+y)f(y)) = cos2π λ,x h i N |λX|2=E is the two-point function of our random process (which is translation invari- ant). This formula shows that one should single out points x Td where ∈ u(x) is close to 1 (clearly u(x) 1). We will show (see section 6.3) that | | | | ≤ the total contribution to the integral near such (suitably defined)“singular” points is bounded by O( u(x)4dx). Td Outside of these “singular” points, we may expand in a Taylor series R (1 u2) 1/2 = 1+ 1u2 +O(u4). The constant term 1 corresponds to the − − 2 square of the expectation and thus we will get 1 Var( ) = u(x)2dx+O u(x)4dx . L 4π Td Td Z (cid:18)Z (cid:19) Thesecondmomentofuisimmediately seentoequal u(x)2dx = 1/ , Td N and it is easily seen that the fourth moment of u is at most 1/ . Thus we R N get an upper bound Var( )= O(1/ ) (in any dimension d 2). To obtain L N ≥ Theorem 1.1 one needs to show that the fourth moment of u is negligible relative to 1/ . In dimension d = 2 we have u(x)4dx 1/ 2 by a N Td ≪ N geometric argument due to Zygmund [21]. In dimension d 3, we can show R ≥ that Ed−23+ǫ (1.2) u(x)4dx , ǫ > 0 Td ≪ǫ 2 ∀ Z N which in dimension d 5 suffices because Ed2−1 and so we get a bound ≥ N ≈ of 1/ E1/2 ǫ. − N Alternatively, note that u(x) is itself an eigenfunction of the Laplacian and we want a bound on its L4-norm relative to its L2-norm. In dimension d 5abound(validforanyRiemannianmanifold)duetoSogge[19]suffices ≥ here. Astronger boundfor thetorus, dueto Bourgain [5], willimprove (1.2) for d 7. ≥ 1.6. Acknowledgements. We thank Misha Sodin for several helpful dis- cussions. This work was supported by the Israel Science Foundation (grant No. 925/06). In addition, I.W. was partly supported by SFB 701: Spectral Structuresand Topological Methods in Mathematics, (Bielefeld University). 2. Random eigenfunctions on the torus 2.1. The basic setup. Wewishtoconsidereigenfunctions oftheLaplacian on the standard flat torus: ∆ψ+4π2Eψ = 0 . Thesecanbewrittenaslinearcombinationsofthebasicexponentialse2πi λ,x , h i with λ Zd, λ 2 = E. The dimension of the corresponding eigenspace ∈ | | N LERAY MEASURE FOR NODAL SETS OF EIGENFUNCTIONS ON THE TORUS 5 is simply the number of ways of expressing E as a sum of d integer squares. For d 5 this grows roughly as Ed/2 1 as E . For d 4 the di- − ≥ → ∞ ≤ mension of the eigenspace need not grow with E. In the extreme case d = 2, is given in terms of the prime decomposition of E as follows: N If E = 2α pβj q2γk where p 1 mod 4 and q 3 mod 4 are odd j j k k j ≡ k ≡ primes, α,β ,γ are integers, then = 4 (β +1), and otherwise E is Qj k ≥Q N j j not a sum of two squares and = 0. On average (over integers which are N Q sums of two squares) the dimension is const √logE. · For some of our initial work, throughout sections 4, 5 we will work in § greater generality and instead of eigenspaces we will consider linear spaces = (Λ) spanned by certain sets of exponentials e2πi λ,x with λ Λ Zd. h i E E ∈ ⊂ We take into account the reflection symmetries of the torus by assuming thatthe frequencyset Λis invariant underthegroup of signed permutations W = 1 d S , consisting of coordinate permutations and sign-change d d {± } × of any coordinate, e.g. (λ ,λ ) ( λ ,λ ) (for d = 2). We say that a 1 2 1 2 7→ − non-empty subset Λ Zd is “symmetric” if it is invariant under W , that is d ⊂ invariant under permutations of the coordinates and changing sign of each coordinate, and that 0 / Λ. ∈ The dimension = dim is the number of the frequencies in Λ. Since N E Λ is symmetric and does not contain 0, is even. We write Λ/ to denote N ± representatives of the equivalence class of Λ under λ λ. 7→ − Lemma 2.1. Any set Λ satisfying the symmetry conditions (i.e. invariant w.r.t. coordinate permutations and sign changes), spans Rd. Proof. Otherwise we have a nontrivial linear relation d (2.1) c λ = 0, i i l=1 X valid for all λ Λ. Since Λ is invariant under permutations, we may as- ∈ sume λ = 0. Substituting λ and λ = ( λ , λ , ..., λ ) and subtracting 1 6 ′ − ′1 2 d the equations we obtain 2λ c = 0, which implies c = 0. Repeating the 1 1 1 argument for all c , we get a contradiction. (cid:3) i As a consequence of this lemma, we see that the set L of integer linear Λ combinations of elements of Λ Zd is a sublattice of full rank, and hence ⊆ its dual L = v Rd : λ,v Z, λ Λ ∗Λ { ∈ h i ∈ ∀ ∈ } is also a lattice in Rd (containing Zd). 2.2. A non-degeneracy condition. Assumethatthesetof frequencies Λ, which is assumed to be “symmetric”, further satisfies the following “non- degeneracy” condition: (2.2) λ Λ with λ = λ and λ ,λ = 0 . 1 2 1 2 ∃ ∈ 6 ± 6 By the symmetry of the set Λ, condition (2.2) is equivalent to requiringthat for every i = j, there is λ Λ with λ = λ and λ ,λ = 0. i j i j 6 ∈ 6 ± 6 In the case of eigenfunctions of the Laplacian, where Λ = λ Zd : { ∈ λ 2 = E , the non-degeneracy condition (2.2) holds as soon as = #Λ | | } N is sufficiently large, in fact if > 3d. This is because any λ where there N 6 FERENCORAVECZ,ZEE´VRUDNICKANDIGORWIGMAN are no distinct indices i = j with λ ,λ = 0, λ = λ must be in the i j i j 6 6 6 ± W -orbit of a vector of the form λ(j,r) = (r,r,...,r,0,...,0) with the first d j coordinates equal to r > 0 and the remaining d j coordinates equal to − zero, and E = jr2 (so r is determined uniquely by E and 0 j d). The ≤ ≤ number of elements in the W -orbit of λ(j,r) is d 2j and summing over all d j 0 j d gives at most 3d possibilities. ≤ ≤ (cid:0) (cid:1) 2.3. Gaussian ensembles. For any symmetric set of frequencies Λ Zd, ⊂ we define an ensemble of Gaussian random functions f by ∈E 1 (2.3) f(x)= b cos2πi λ,x c sin2πi λ,x λ λ √2 h i− h i N λX∈Λ with b ,c N(0,1) real Gaussians of zero mean and variance 1 which λ λ ∼ are independent save for the relations b = b , c = c . Thus we can λ λ λ λ − − − rewrite 2 f(x)= b cos2πi λ,x c sin2πi λ,x λ λ h i− h i rN λ Λ/ ∈X± where now only independent random variables appear. Alternatively,wemayidentify = R bytakingcoordinatesZ = (b , c ) and putting the Gaussian probabEil∼ity mNeasure λ λ λ∈Λ/± dµ (Z) = 1 e−(b2λ+c2λ)/2dbλdcλ . N (2π) /2 N λ Λ/ ∈Y± We define a set by B 1 = w Rd : λ,w Z λ Λ or λ,w +Z λ Λ . B { ∈ h i ∈ ∀ ∈ h i ∈ 2 ∀ ∈ } Then clearly 1L L and so the projection of on the torus Td = 2 ∗Λ ⊆ B ⊆ ∗Λ B Rd/Zd is finite. Note that if x y , then for all f , − ∈ B ∈ E f(y) = f(x), and f(y)= f(y). ± ∇ ±∇ For a= (a ,a ) R2, let 1 2 ∈ a = f :f(x) = a ,f(y) = a . Px,y { ∈ E 1 2} If x y / then this is an affine hyperplane of codimension two in . If − ∈ B E x y then this is either empty or a hyperplane of codimension one in − ∈ B . E We define the two-point function of our ensemble as u(x,y) = E(f(x)f(y)). A simple computation shows that u(x,y) depends only on the difference x y, in fact u(x,y) = u(x y) where − − 1 u(z) = cos2π λ,z . h i N λ Λ X∈ Lemma 2.2. u(x) = 1 if and only if x . ± ∈ B LERAY MEASURE FOR NODAL SETS OF EIGENFUNCTIONS ON THE TORUS 7 Proof. If x then cos2π λ,x are all equal, to either +1 or 1 and hence ∈ B h i − u(x) = 1. On the other hand, since cos2π λ,x 1, if u(x) = 1 then ± | h i| ≤ ± all the cosines cos2π λ,x have the same value, which is either +1 or 1, h i − and this forces either λ,x Z for all λ Λ, or λ,x 1+Z for all λ Λ, that is x . h i ∈ ∈ h i ∈ 2 ∈ (cid:3) ∈B 2.4. The singular set. We define the set of singular functions to be Sing := f : x Td, f(x)= 0 and ( f)(x)=~0 . { ∈ E ∃ ∈ ∇ } Lemma 2.3. The set Sing has codimension at least 1 in . E Proof. Define ψ :Td Rd R ×E → × (x, f) ( f(x), f(x)), 7→ ∇ Denoting π : Td R R the projection to the second factor, we 2 N N × → have Sing = π (ψ 1( 0 0 )). 2 − { }×{ } WeprovethattheJacobianofψhasmaximalrankeverywhere,andtherefore ψ 1( 0 0 ) isasmoothmanifoldofcodimensiond+1. Itwillthenfollow − { }×{ } that Sing R has codimension 1. N ⊂ ≥ The (d+1) (d+ ) Jacobian matrix is × N 2π 2 A(x) Dψ(x) = ∗ − N ,  q2 B(x)  ∗ N  q  where A(x) is a d matrix defined by ×N A(x) = (sin2π λ,x ~λ, cos2π λ,x ~λ) , h i h i (cid:18) (cid:19)λ Λ/ ∈ ± and B(x) is a 1 matrix defined by ×N B(x)= (cos2π λ,x , sin2π λ,x ) . h i − h i (cid:18) (cid:19)λ Λ/ ∈ ± A Thus we want the (d+1) matrix to have rank d+1. However, ×N B (cid:18) (cid:19) ordering the vectors ~λ(j) Λ/ , it is a product of ∈ ± ~0 ~λ(1) ~0 ~λ(2) ... , 1 0 1 0 ... (cid:18) (cid:19) which is of rank d+1 by lemma 2.1 and cos2π λ(1),x sin2π λ(1),x 0 0 ... h i − h i sin2π λ(1),x cos2π λ(1),x 0 0 ...  h i h i  0 0 cos2π λ(2),x sin2π λ(2),x ... h i − h i  0 0 sin2π λ(2),x cos2π λ(2),x ...  h i h i   ..   .   which is nonsingular. This immediately implies the result. (cid:3) The following is an immediate 8 FERENCORAVECZ,ZEE´VRUDNICKANDIGORWIGMAN Corollary 2.4. The set Sing has measure zero in . E 3. The Leray measure We continue with our previous setting, that is Λ Zd is a symmetric, ⊂ non-degeneratesetoffrequencies. WewishtodefinetheLeraymeasure (f) L for f by the limit ∈E 1 (f)= lim meas x: f(x) <ǫ . L ǫ 02ǫ { | | } → It is well known that the limit exists for any nonsingularf (see [10, Chapter III], [16, 3.3]), and that in fact § dσ(x) (f)= L f(x) Z{x:f(x)=0} |∇ | where dσ(x) is the induced hypersurface measure. We willneed to know morerefinedinformation abouttheapproach tothe limit in the definition. For ǫ > 0, set 1 (f) := meas x : f(x) < ǫ . ǫ L 2ǫ { | | } so that (f)= lim (f). ǫ 0 ǫ L → L For α> 0, β > 0 let (α, β) = f : f(x) α f(x) >β . E { ∈ E | | ≤ ⇒ |∇ | } The sets (α, β) are open, and have the monotonicity property E α > α (α ,β) (α ,β) 1 2 1 2 ⇒ E ⊆ E and β > β (α,β ) (α,β ) . 1 2 1 2 ⇒ E ⊆ E Moreover, for any sequence α ,β 0 we have n n → Sing = (α , β ) . n n E \ E n [ Lemma 3.1. For f (α, β) and 0 < ǫ < α, we have ∈ E d3/2 (f)< 2 E ǫ max L β p where E = max λ 2 :λ Λ . max {| | ∈ } We will first treat the one variable (d = 1) case and state it as a separate lemma (cf [11, Lemma 2]): Lemma 3.2. Let g(t) be a trigonometric polynomial of degree at most M so that there are α > 0, β > 0 such that g (t) > β whenever g(t) < α. ′ | | | | Then for all 0 < ǫ < α we have 1 2M meas t : g(t) < ǫ < . 2ǫ { | | } β LERAY MEASURE FOR NODAL SETS OF EIGENFUNCTIONS ON THE TORUS 9 Proof. Decompose the open set t : g(t) < ǫ as a disjoint union of open { | | } intervals (a ,b ) (with a < b ) and such that on each such interval, g has k k k k ′ constant sign, thatis either g > β or g < β. We willshow that thelength ′ ′ − b a of each such interval is at most 2ǫ/β and that there are at most 2M k k − such intervals. Suppose that on (a ,b ), g > β; then g is increasing, and g(a ) = ǫ, k k ′ k − g(b )= +ǫ. Then the length of the interval is k bk g (t) 1 bk ′ b a = dt < g (t)dt k k ′ − g (t) β Zak ′ Zak g(b ) g(a ) 2ǫ k k = − = . β β Likewise, if g < β on (a ,b ) then g(a )= +ǫ, g(b ) = ǫ, and ′ k k k k − − bk g (t) 1 bk ′ bk ak = − dt < g′(t)dt − g (t) β − Zak − ′ Zak g(a ) g(b ) 2ǫ k k = − = β β as required. In both cases, each interval has an endpoint where g(t) = +ǫ, and hence thenumberofsuchintervals isboundedbythenumberof solutionsof g(t) = +ǫ which is at most 2M since g is a trigonometric polynomial of degree at most M. (cid:3) We now prove Lemma 3.1 by reduction to the case d = 1: Proof. Decompose the set x: f(x) < ǫ as a union d W where { | | } ∪j=1 j ∂f ∂f W = y : f(y) < ǫ, (y) (y) k = j j { | | |∂x | ≥ |∂x | ∀ 6 } j k and it suffices to show that √d meas(W ) < 2ǫ 4 E . j max β p For simplicity we fix j = 1. On W , we have 1 ∂f β (y) > |∂x1 | √d since f(y) < ǫ < α implies (recall f (α,β)) | | ∈ E d ∂f ∂f β2 < f(y)2 = (y)2 d (y)2 . |∇ | |∂x | ≤ |∂x | k 1 k=1 X For y Td 1 set − ∈ I(y) = t T1 : (t,y) W 1 { ∈ ∈ } which is a subset of T1. Then slice-integration gives meas(W ) = meas(I(y))dy 1 Td−1 Z 10 FERENCORAVECZ,ZEE´VRUDNICKANDIGORWIGMAN and so it suffices to show √d meas(I(y)) < 2ǫ 4 E . max β p Now on I(y), the one-variable trigonometric polynomial g(t) := f(t,y) satisfies g(t) = f(t,y) < ǫ, and | | | | ∂f β g (t) = (t,y) > . ′ | | |∂x1 | √d Moreover g(t) is of degree at most √E because max d f(t,y)= a e(λ t+ λ y ) λ 1 j j λ Λ j=2 X∈ X and for all frequencies in the sum we have λ2 λ 2 E . Thus by 1 ≤ | | ≤ max Lemma 3.2 we find that meas(I(y)) < 2ǫ√d2√E as required. (cid:3) β max 4. The expected value of L In this section, we give a formula for the expected value of (f): L Theorem 4.1. Suppose that Λ issymmetric and satisfies the nondegeneracy condition (2.2). Then the Leray measure (f) is integrable (with respect to L the Gaussian measure), and 1 (4.1) E( )= . L √2π 4.1. A formal treatment. To compute the expectation of (f), we for- L mally write it as (f)= δ(f(x))dx L Td Z and hence formally E( (f))= E( δ(f(x))dx) = E(δ(f(x))dx L Td Td Z Z Now for each fixed x Td, the random variable f(x) is a sum of Gaussians ∈ hence is itself a Gaussian whose mean is zero and variance is computed to be unity. Hence the expected value E(δ(f(x)) should be E(δ(f(x)) = ∞ δ(a)e a2/2 da = 1 − √2π √2π Z−∞ which gives the result E( ) = 1/√2π. Justifying this simple manipulation L in a rigorous fashion turns out to be rather tedious will be done below, with some parts relegated to an appendix.