Expected nodal volume for non-Gaussian random band-limited functions
EEXPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOMBAND-LIMITED FUNCTIONS
ZAKHAR KABLUCHKO, ANDREA SARTORI, AND IGOR WIGMAN
Abstract.
The asymptotic law for the expected nodal volume of random non-Gaussianmonochromatic band-limited functions is determined in vast generality. Our methodscombine microlocal analytic techniques and modern probability theory. A particularlychallenging obstacle needed to overcome is the possible concentration of nodal volumeon a small proportion of the manifold, requiring solutions in both disciplines. As for thefine aspects of the distribution of nodal volume, such as its variance, it is expected thatthe non-Gaussian monochromatic functions behave qualitatively differently compared totheir Gaussian counterpart, with some conjectures been put forward. Introduction
In the recent years a lot of effort has been put into understanding the geometry ofLaplace eigenfunctions on smooth manifolds. Let (
M, g ) be a smooth compact Riemann-ian manifold of dimension n , and ∆ = ∆ g the Laplace-Beltrami operator on M . Denote { λ i } i ≥ to be the (purely discrete) spectrum of ∆, with the corresponding Laplace eigen-functions φ i satisfying ∆ φ i + λ i φ i = 0 . An important qualitative descriptor of the geometry of φ i is its nodal set φ − i (0), and,in particular, the nodal volume V ( φ i ) = H n − ( φ − i (0)), that is the ( n − φ − i (0).The highly influential Yau’s conjecture [56] asserts that, in the high energy limit λ i →∞ , the nodal volume of φ i is commensurable with λ i : there exists constants C M > c M > c M · λ i ≤ V ( φ i ) ≤ C M · λ i . Yau’s conjecture was established for the real analytic manifolds [9, 10, 17], whereas, morerecently, the optimal lower bound and polynomial upper bound were proved [29, 30, 31]in the smooth case.In his seminal work [6] Berry proposed to compare the (deterministic) Laplace eigen-functions on manifolds, whose geodesic flow is ergodic, to the random monochromaticisotropic waves, that is, a Gaussian stationary isotropic random field F µ : R n → R , whosespectral measure µ is the hypersurface measure on the sphere S n − ⊆ R n , normalized byunit total volume. Equivalently, F µ ( · ) is uniquely defined via its covariance function K ∞ ( x, y ) := E [ F µ ( x ) · F µ ( y )] = (cid:90) S n − e πi (cid:104) x − y,ξ (cid:105) dµ ( ξ ) (1.1)For example, in 2 d , the covariance function of F µ : R → R is given by E [ F µ ( x ) · F µ ( y )] = J ( | x − y | ) , a r X i v : . [ m a t h . P R ] F e b XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 2 with J the Bessel J function of order 0. Berry’s conjecture should be understood in somerandom sense, e.g. when averaged over the energy level. Alternatively, one can considersome random ensemble of eigenfunctions or their random linear combination, Gaussianor non-Gaussian.A concrete ensemble of the said type is that of band-limited functions [50] f T ( x ) = f ( x ) = v ( T ) − / (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] a i φ i ( x ) , (1.2)where a i are centred unit variance i.i.d. random variables, Gaussian or non-Gaussian, T →∞ is the spectral parameter , and the summation is over the energy window [ T − ρ ( T ) , T ]of width ρ = ρ ( T ) ≥
1. The convenience pre-factor, which has no impact on the nodalstructure of f T ( · ), v ( T ) := (2 π ) n ω n · Vol( M ) ρ ( T ) T n − = c M ρ ( T ) T n − , (1.3)where ω n is the volume of the unit ball in R n , represents the asymptotic, as T → ∞ ,number of summands in (1.2) so that f T ( x ) is of asymptotic unit variance at each x ∈ M .Regardless of whether or not f T ( · ) in (1.2) is Gaussian, its covariance kernel is thefunction K T : M × M → R given by K T ( x, y ) := E [ f T ( x ) · f T ( y )] = 1 v ( T ) (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] φ i ( x ) · φ i ( y ) , (1.4)coinciding with the spectral projector in L ( M ) onto the subspace spanned by the eigen-functions { φ i } λ i ∈ [ T − ρ ( T ) ,T ] (recall that a i are unit variance). In what follows we will focuson the most interesting (and, in some aspects, most difficult) monochromatic regime ρ ( T ) = o T →∞ ( T ) (“short energy window”). Our main result (Theorem 1.1) along withall our arguments remain valid for the case ρ ∼ T →∞ α · T, with some α ∈ [0 , M (explicated below), and assum-ing, as we did, that ρ ( T ) T −→ T →∞ √ T , is asymptotic to (1.1), around every (or almost every) reference point x , in the fol-lowing sense. Let exp x : T x M → M be the exponential map, that is a bijection betweena ball B ( r ) ⊆ R n centred at 0 ∈ R n and some neighbourhood in M of x , with r > M , independent of x ∈ M . Then, for every R > K T (exp x ( x/T ) , exp x ( y/T )) −→ T →∞ K ∞ ( x, y ) , (1.5)uniformly for (cid:107) x (cid:107) , (cid:107) y (cid:107) ≤ R , with K ∞ ( · , · ) as in (1.1), with the convergence (1.5) holdingtogether with an arbitrary number of derivatives [11, 12, 48]. Since in the Gaussian case,the finite-dimensional distributions are determined by the covariances, if f T ( · ) is Gaussian,the convergence (1.5) readily implies that the law of f T locally converges to that of F µ .For the non-Gaussian case, a significant proportion of our argument below will exploit the If M is periodic and ρ ( T ) = O (1) we assume that ρ ( T ) is sufficiently large, depending on M , so thatto avoid trivialities. Alternatively, one may assume that T belongs to a particular subsequence, see (3.2)below. XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 3 the increasing number of summands in (1.2) to apply the Central Limit Theorem togetherwith the convergence (1.5) to yield the local convergence of the law of f T to that of F µ .For the linear combinations (1.2) of Laplace eigenfunctions on real analytic M , thedeterministic upper bound analogue V ( f T ) ≤ C M · T of Yau’s conjecture remains valid [36, Section 14]. The principal result of this manuscriptdetermines the precise asymptotic growth of the expected nodal volume of monochromaticrandom band-limited functions on generic real analytic manifolds with no boundary, underthe mere assumptions that a n have finite second moment: Theorem 1.1.
Let ( M, g ) be a real analytic compact n -manifold with empty boundary,assume that ρ ( T ) = o T →∞ ( T ) , and either n ≥ and ρ ( T ) ≥ , or n = 2 and ρ ( T ) ≥ T log T ,and let f T ( · ) be the band-limited functions (1.2) with a n centred i.i.d. so that E [ a n ] = 1 .Then E [ V ( f T )] = Vol( M ) (cid:18) πn (cid:19) / Γ (cid:0) n +12 (cid:1) Γ (cid:0) n (cid:1) T + o T →∞ ( T ) . (1.6) Some conventions.
We write A (cid:46) B to designate the existence of some constant C > A ≤ CB . We also write C, c >
M, g ), but,since (
M, g ) is fixed, we suppress this dependence in the notation. We also write B ( x, r )for the (Euclidean) ball centred at x of radius r > B g ( · ) for the geodesic ball and, forbrevity, we write B = B (0 , ⊂ R n . Given a ball B and some r >
0, we write B for itsclosure and rB for the concentric ball with r -times the same radius. Moreover, we usethe multi-index notation D α = ∂ α x ...∂ α x where α = ( α , ..., α n ) and | α | = α + ... + α n .Furthermore, given a ( C ) function g : B ( x, r ) → R and some r >
0, we let V ( g, B ( x, r )) = H n − { x ∈ B ( x, r ) : g ( x ) = 0 } be the nodal volume of g in B ( x, r ). Since it will be often useful to change scales, we alsowrite g r ( · ) = g ( r · ) and notice that V ( g r , B (1)) r n − = V ( g, B ( r )) . Finally, we denote by Ω the abstract probability space where every random object isdefined.
Acknowledgements.
We would like to thank S. Zelditch for useful discussions, and,in particular, for sharing with us his unpublished results on self-focal points on analyticmanifolds, demonstrating that the class of real analytic manifolds, for which Theorem1.1 applies, is unrestricted, see the discussion in section 2.3. A. Sartori was supportedby the Engineering and Physical Sciences Research Council [EP/L015234/1], ISF Grant1903/18 and the IBSF Start up Grant no. 2018341. Z. Kabluchko was supported bythe German Research Foundation under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics M¨unster: Dynamics - Geometry - Structure.2.
Discussion
Survey of non-Gaussian literature.
To our best knowledge, Theorem 1.1 is thefirst universality result applicable in the asserted vastly general scenario, in terms of boththe underlying manifold M and the random coefficients { a i } . Our approach is based on ablend of microlocal analytic techniques, missing from the existing non-Gaussian literature, XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 4 and purely probabilistic methods. The closest analogue to Theorem 1.1 we are awareof in the existing literature is [1], dealing with 2d random non-Gaussian trigonometricpolynomials: these are related to the random band-limited Laplace eigenfunctions on thestandard 2d torus, corresponding to the long energy window ρ ( T ) = T (here, the energiesordering is somewhat different, to allow for separation of variables). The asymptoticsfor the expected nodal length was asserted for centred unit variance random variables,in perfect harmony to (1.6) (though with a different leading constant, a by-product of anon-monochromatic scaling limit).Even though we didn’t meticulously validate all the details, we believe that their argu-ments translate verbatim for the “pure” 2d toral Laplace eigenfunctions g n ( x ) = (cid:88) µ ∈ Z (cid:107) µ (cid:107) n a µ · e ( (cid:104) µ, x (cid:105) ) , (2.1)where the a µ are i.i.d., save for the relation a − µ = a µ making g n real-valued, and thesummation on the r.h.s. of (2.1) is w.r.t. to all standard lattice points lying on the radius- √ n centred circle. In the Gaussian context the g n are usually referred to as “arithmeticrandom waves” (ARW), see e.g. [24, 43, 47]; they are the band-limited functions forthe standard flat torus corresponding to the “short energy window” ρ ( T ) = 1 (in fact,in this case, the energy window width could be made infinitesimal). Other than theresult for 2d random trigonometric polynomials all the literature concerning real zerosof non-Gaussian ensembles is 1-dimensional in essence: real zeros of random algebraicpolynomials or Taylor series, see e.g. [22, 23, 42] and the references therein, randomtrigonometric polynomials on the circle [3], and the restrictions of 2d random toral Laplaceeigenfunctions (2.1) to a smooth curve [14].2.2. Gaussian vs. non-Gaussian monochromatic functions: cases of study.
Un-like the non-Gaussian state of art concerning the zeros of the band-limited functions,the Gaussian literature is vast and rapidly expanding, thanks to the powerful Kac-Ricemethod tailored to this case, at times, combined with the Wiener chaos expansion. Herethe literature varies from the very precise and detailed results concerning the zero volumedistribution (its expectation, variance and limit law), restricted to some particularly im-portant ensembles, such as random spherical harmonics [35, 55] or the arithmetic randomwaves [24, 34], to somewhat less detailed results, but of far more general nature [13, 59],to almost sure asymptotic result [19] w.r.t. a randomly independently drawn sequence offunctions { f T } T .It is plausible, if not likely, that, under a slightly more restrictive assumptions on therandom variables, our techniques yield a power saving upper bound for the nodal lengthvariance of the type Var (cid:18) f T T (cid:19) = O (cid:0) T − δ (cid:1) for some δ >
0, but certainly not a precise asymptotic law for the variance, even for theparticular cases of non-Gaussian random spherical harmonics or the non-Gaussian Arith-metic Random Waves. In the Gaussian case even some important non-local properties ofthe nodal set were addressed: the expected number of nodal components [38, 39], theirfluctuations [16, 40], their fine topology and geometry, and their relative position [4, 50].
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 5
The aforementioned random ensemble of Gaussian spherical harmonics is the sequenceof functions f (cid:96) : S → R , (cid:96) ≥
1, where f (cid:96) ( x ) = 1 √ (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) a (cid:96),m Y (cid:96),m ( x ) , (2.2)with { Y (cid:96),m } − (cid:96) ≤ m ≤ (cid:96) the standard basis of degree- (cid:96) spherical harmonics, and a (cid:96),m i.i.d.standard Gaussian random variables. An application of the Kac-Rice formula yields [5]the expected nodal length of f (cid:96) ( · ) to be given precisely by E [ V ( f (cid:96) )] = √ π · (cid:112) (cid:96) ( (cid:96) + 1) ∼ √ π(cid:96), whereas a significantly heavier machinery, also appealing to the Kac-Rice method, yields[55] a precise asymptotic law Var( V ( f (cid:96) )) ∼ (cid:96) →∞
132 log (cid:96), smaller than what would have been thought the natural scaling ≈ c · (cid:96) would be (“Berry’scancellation phenomenon”).For the non-Gaussian random spherical harmonics, Theorem 1.1 is not directly appli-cable, because of the extra condition ρ ( T ) ≥ T log T in 2d. However, in Appendix A belowwe were able to extend the validity of Theorem 1.1 to this important ensemble, at leastfor Bernoulli random variables (see Theorem A.1). In light of the non-universality resultof [3], it is not unlikely that in the non-Gaussian case (i.e. the a (cid:96),m are centred unitvariance i.i.d. random variable), the variance satisfies the 2-term asymptoticsVar( V ( f (cid:96) )) = c · (cid:96) + c · log (cid:96) + O (1) , with c , c depending on the law of a (cid:96),m and c vanishing for a peculiar family of distri-butions, including the Gaussian one. It seems less likely, though conceivable , that c ≡ E [ V ( g n )] = π √ · √ n , whereas the variance is asymptoticto Var( V ( g n )) ∼ π b n · nr ( n ) , where r ( n ) is the number of summands in (2.1). Here the numbers b n are genuinelyfluctuating in [1 / , / nr ( n ) “miraculously” cancelling out precisely (“arithmetic Berry’s cancellation”).Using the same reasoning as for the spherical harmonics, for the non-Gaussian case(i.e. a µ are centred unit variance i.i.d. random variables), it is expected that the 2-termasymptotics Var( V ( g n )) ∼ (cid:101) c nr ( n ) + (cid:101) c nr ( n ) holds with (cid:101) c , (cid:101) c possibly depending on both the law of a µ and the angular distributionof the lattice points { µ } in (2.1), with (cid:101) c vanishing for a µ a peculiar class of distributionlaws, including the Gaussian (whence (cid:101) c vanishes independent of the angular distributionof the lattice points { µ } ). The dependence of (cid:101) c and (cid:101) c on both the distribution law of a µ and the angular distribution of { µ } is of interest, in particular, whether the vanishingof (cid:101) c depends on the angular distribution of { µ } at all (which is not the case if a µ is XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 6
Gaussian). Again, it is conceivable that (cid:101) c ≡
0. We leave all of the above questions to beaddressed elsewhere.2.3.
Self-focal points.
An earlier version of this manuscript contained a version of Theo-rem 1.1, applicable under a seemingly somewhat restrictive (though very mild) assumptionon M , rather than its mere analyticity, concerning its so-called self-focal points. It waspointed to us by S. Zelditch, that the extra assumption is redundant, as explained below,after a few necessary definitions. Definition 2.1.
Let (
M, g ) be a smooth compact manifold without boundaries, S ∗ M thecotangent sphere bundle on M , and G t : S ∗ M → S ∗ M the geodesic flow on M .(1) The set of loop directions based at x is L x = { ξ ∈ S ∗ x M : ∃ t > . exp x ( tξ ) = x } . (2) The set of closed geodesics based at x is CL x = { ξ ∈ S ∗ x M : ∃ t > . G t ( x, ξ ) = ( x, ξ ) } . (3) A point x ∈ M is self-focal , if |L x | >
0, where | · | is the natural measure on S ∗ x induced by the metric g x ( · , · ).(4) The geodesic flow on M is periodic , if the set of its closed geodesics is of fullLiouville measure in S ∗ M . The geodesic flow on M is aperiodic if the set of itsperiodic closed geodesics is of Liouville measure 0.We observe that for M real analytic, the set of its periodic geodesics, is of either full or0 Liouville measure in S ∗ M (see either [48, Lemma 1.3.8] or Lemma 3.1 below). Hence,in the real analytic case, the geodesic flow on M is either periodic or aperiodic .Originally, Theorem 1.1 assumed that if M is aperiodic, then the set of its self-focalpoints is of measure 0. However, it was demonstrated [57] that if M is aperiodic, then theset of self-points has automatically measure 0, i.e. the said extra assumption is redundant,see also section 3.2 below. 3. Preliminaries
Geodesic flow and the spectrum of √− ∆ on M . Let T ∗ M and S ∗ M be theco-tangent and the co-sphere bundle on M respectively. The geodesic flow G t : T ∗ M → T ∗ M (3.1)is the Hamiltonian flow of the metric norm function H : T ∗ M → R H ( x, ξ ) = n (cid:88) i,j =1 g ij ξ i ξ j , where g = g ij is the metric on M and g ij is its inverse. Since G t is homogeneous, fromnow on, we will consider only its restriction to S ∗ M . We will need the following simplelemma, see also [48, Lemma 1.3.8] Lemma 3.1. If ( M, g ) is a real analytic manifold, then the set of closed geodesics, on theco-sphere bundle equipped with the Liouville measure, has either full measure or measurezero. XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 7
Proof.
Since closed geodesics are defined by G t ( x, ξ ) = ( x, ξ ) , the set of closed geodesic, for fixed t > (cid:3) Lemma 3.1 implies that the geodesic flow, on a real analytic manifold, is either aperiodic if there set of closed geodesics has measure zero, or periodic with (minimal) period
H > G H = id . In the latter case the manifold is also called Zoll . Therefore, we have thefollowing description of the spectrum of √− ∆ of real-analytic ( M, g ), see [60, Theorem8.4] and references therein.Suppose that the geodesic flow on M is aperiodic . The two-term Weyl law of Duistermaat-Guilleimin(-Ivrii) states |{ i > λ i ≤ T }| = C m T n + o ( T n − ) . Now assume that the geodesic flow on M is periodic with period H . Then the spectrumof √ ∆ is a union of clusters of the form C k := (cid:26) πH (cid:18) k + β (cid:19) + µ k i for i = 1 , ..., d k (cid:27) k = 1 , ..., where µ k i = O ( k − ) uniformly for all i , d k is a polynomial in k of degree n − β isthe Morse index of M . In particular, in order to avoid trivial summation in (1.2), if thegeodesic flow is periodic and ρ ( T ) = O (1), we will assume that either T = T ( k ) is of theform T = 2 πH (cid:18) k + β (cid:19) + 1 , (3.2)or, alternatively, that ρ ( T ) is sufficiently large.We will need the following Lemma, see [59, Proposition 2.3 (A)]. Lemma 3.2.
Let ( M, g ) be a real analytic compact manifold without boundary of dimen-sion n . Then, uniformly for all x ∈ M , we have (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] | φ i ( x ) | = c M ρ ( T ) T n − (1 + o T →∞ (1)) . Local Weyl’s law.
To state the main result of this section, we first need to introducesome notation. Let x ∈ M and let F x be f rescaled to the ball B g ( x, /T ) in normalcoordinates. More precisely, following [39], we define: F T,x ( y ) = F x ( y ) = f (exp x ( y/T )) (3.3)for y ∈ B (0 ,
1) =: B ⊂ R n , where exp x : R n ∼ = T x M → M is the exponential map.Notice that, in the definition of the exponential map, we have tacitly identified, via anEuclidean isometry, R n with T x M . Moreover, we observe that, since ( M, g ) is analytic,the injectivity radius of M is uniformly bounded from below [15], thus, from now on, weassume that 1 /T is smaller than the injectivity radius so that the exponential map is adiffeomorphism. Furthermore, thanks to [39, Section 8.1.2] due to Nazarov and Sodin,see also [46, Section 2], the map:( x, ω ) ∈ M × Ω → F x ( ω, · ) ∈ C ∞ ( B )is measurable.The main result of this section is the following: XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 8
Proposition 3.3.
Let F x be as in (3.3) and M be a compact, real-analytic manifold withempty boundary. Then, for x ∈ M outside of a measure set, one has E [ F x ( y ) F x ( y (cid:48) )] = (cid:90) | ξ | =1 e ( (cid:104) y − y (cid:48) , ξ (cid:105) ) dξ + o (1) = J Λ ( | y − y (cid:48) | ) | y − y (cid:48) | Λ + o T →∞ (1) where Λ = ( n − / and J Λ ( · ) is the Λ -th Bessel function, uniformly for all y, y (cid:48) ∈ B .Moreover, we can also differentiate both sides any arbitrary finite number of times, thatis E [ D α F x ( y ) D α (cid:48) F x ( y (cid:48) )] = ( − | α (cid:48) | (2 πi ) | α | + | α (cid:48) | (cid:90) | ξ | =1 ξ α + α (cid:48) e ( (cid:104) y − y (cid:48) , ξ (cid:105) ) dξ + o T →∞ (1) where α, α (cid:48) are multi-indices, and ξ α = ( ξ α , ..., ξ α n n ) . In order to prove Proposition 3.3, we will need the following fact communicated to usby S. Zelditch [57]:
Lemma 3.4.
Let M be a compact, real-analytic manifold with empty boundary. If thegeodesic flow on M is aperiodic then the set of self-focal points, as in Definition 2.1, haszero volume. In light of Lemma 3.4, in order to prove Proposition 3.3, it is sufficient to prove thefollowing:
Proposition 3.5.
Let ( M, g ) be a compact, real-analytic manifold with empty boundary.Suppose that the geodesic flow on M is periodic or x ∈ M is not a self-focal point, as inDefinition 2.1, then sup y,y (cid:48) ∈ B g ( x, /T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] φ i ( y ) φ i ( y (cid:48) ) − (2 π ) − n c M T n J Υ( T ) ( T d g ( y, y (cid:48) )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o ( T n − ) (3.4) where d g ( y, y (cid:48) ) is the geodesic distance between y, y (cid:48) , c M > is given in (1.3) , Υ( T ) =1 − ρ ( T ) T and J Υ( T ) ( w ) = (cid:90) Υ( T ) ≤| ξ |≤ e ( (cid:104) w, ξ (cid:105) ) dξ. (3.5) Moreover, we can also differentiate both sides of (3.4) any arbitrary finite number oftimes, that is, sup y,y (cid:48) ∈ B g ( x, /T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:80) λ i ∈ [ T − ρ ( T ) ,T ] D αy φ i ( y ) D α (cid:48) y (cid:48) φ i ( y (cid:48) ) − c M T n D αy D α (cid:48) y (cid:48) J Υ( T ) ( T d g ( y,y (cid:48) ))(2 π ) n (cid:12)(cid:12)(cid:12)(cid:12) T | α | + | α (cid:48) | = o ( T n − ) where α, α (cid:48) are multi-indices, and ξ α = ( ξ α , ..., ξ α n n ) and the derivatives are understoodafter taking normal coordinates around the point x . The proof of Proposition 3.5 follows directly from the following two lemmas. In the casethat the geodesic flow is periodic, we have a full asymptotic expansion for the spectralprojector kernel [58], see also [59]. In particular, we have the following:
Lemma 3.6 (Zelditch) . Let ( M, g ) be a compact, real-analytic manifold with empty bound-ary. Suppose that M is Zoll, then the conclusion of Proposition 3.5 holds. The second lemma is borrowed from Canzani-Hanin [11, 12], see also the precedingwork of Safarov [49]:
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 9
Lemma 3.7.
Let ( M, g ) be a compact, real-analytic manifold with empty boundary. Sup-pose that x ∈ M is not self-focal, as in Definition 2.1, then the conclusion of Proposition3.5 holds. We are finally ready to prove Proposition 3.3:
Proof of Proposition 3.3.
First we observe that, thanks to Lemma 3.4, under the assump-tions of Proposition 3.3, the conclusion of Proposition 3.5 holds for almost all x ∈ M .Indeed, thanks to Lemma 3.1, we may assume that the geodesic flow on M is either aperi-odic or periodic. In the latter case, the conclusion of Proposition 3.5 holds for all x ∈ M .In the former case, Lemma 3.7 holds for almost all x ∈ M .Hence, we are left with showing how the conclusion of Proposition 3.5 implies theconclusion of Proposition 3.3. Since ρ ( T ) = o ( T ), re-writing the integral in (3.5) thespherical coordinates, we have (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] φ i ( y (cid:48) ) φ i ( y ) = (2 π ) − n c M ρ ( T ) T n − (cid:90) | ξ | =1 e ( (cid:104) T d g ( y (cid:48) , y ) , ξ (cid:105) ) dξ + O (cid:0) ρ ( T ) T n − d g ( y (cid:48) , y ) (cid:1) + o ( T n − )= c M ρ ( T ) T n − J Λ ( | T d g ( x, y ) | ) | T d g ( y (cid:48) , y ) | Λ + O (cid:0) ρ ( T ) T n − d g ( y (cid:48) , y ) (cid:1) + o ( T n − ) , (3.6)where Λ = ( n − /
2, and Proposition 3.3 follows. (cid:3)
Sogge’s bound.
Let φ i be an eigenfunction with eigenvalue λ i . Then, we have thefollowing estimate on the L p norms of φ i [53], see also [60, Theorem 10.1]: || φ i || L p (cid:28) λ σ ( p ) i || φ i || L (3.7)where σ ( p ) = n − (cid:16) − p (cid:17) < p ≤ n +1) n − n (cid:16) − p (cid:17) − / p ≥ n +1) n − . Asymptotic Gaussianity
Before stating the main result of this section, we need to introduce some notation. Wedenote F µ to be the monochromatic isotropic Gaussian field on B ⊂ R n with spectralmeasure µ , the Lebesgue measure on the n − S n − . Equivalently, F µ has the covariance function E [ F µ ( y ) · F µ ( y (cid:48) )] = (cid:90) | ξ | =1 e ( (cid:104) y − y (cid:48) , ξ (cid:105) ) dξ = J Λ ( | y − y (cid:48) | ) | y − y (cid:48) | Λ , where Λ = ( n − /
2. In what follows we will use the shorthands V ( F x ) := V (cid:18) F x , B (cid:19) and V ( F µ ) := V (cid:18) F µ , B (cid:19) . The aim of this section is to prove the following proposition:
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 10
Proposition 4.1.
Let F x be as in (3.3) and F µ be as above. Under the assumptionsof Theorem 1.1, there exists a subset A ⊂ M of volume at most O (log T /T ) such that,uniformly for all x ∈ M \ A , we have V ( F x ) d −→ V ( F µ ) T → ∞ , the convergence in distribution. To ease the exposition, we split the proof of Proposition 4.1 into a series of steps: firstwe prove that F x , as in (3.3), converges to F µ in distribution in the appropriate spaceof functions, then we deduce Proposition 4.1 using a stability notion for the nodal setintroduced in [39]. Before beginning the proof, we need the following consequence ofSogge’s bound (3.7).4.1. Consequence of Sogge’s bound.
In this section we prove the following conse-quence of (3.7):
Lemma 4.2.
Let K = K ≥ be some parameter which may depend on T , and v ( T ) beas in (1.3) . Then there exists a subset A ⊂ M of volume at most O ( K n +1 n − T − ) such that(1) uniformly for all x ∈ M \ A we have max λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | φ i | (cid:46) K − v ( T ) / (2) uniformly for all x ∈ M \ A we have max λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | T − ∇ φ i | (cid:46) K − v ( T ) / To state the next result, given a Laplace eigenfunction φ i , we denote by φ i,x the restric-tion of φ i to B g ( x, /T ) via the exponential map, that is φ i,x ( y ) = φ i (exp x ( y/T )) , for y ∈ B (0 ,
4) (here we tacitly assume that T is sufficiently large so that 4 /T is less thanthe injectivity radius), see also the definition of F x at the beginning of section 3.2. Withthis notation in mind, we prove the following, see also [51]: Lemma 4.3.
Let T ≥ and φ i let be a Laplace eigenfunction with λ i ∈ [ T − ρ ( T ) , T ] ,then(1) for all x ∈ M , we have sup B g ( x, /T ) | φ i | (cid:46) (cid:90) B (0 , | φ i,x ( y ) | dy. (2) for all x ∈ M , we have sup B g ( x, /T ) | T − ∇ φ i | (cid:46) (cid:90) B (0 , | φ i,x ( y ) | dy. Proof.
Given φ i , let us consider the function h ( x, t ) = φ i ( x ) e λ i t defined on M × [ − , h T ( · ) = h ( T − · ). Then, since the supremum norm is scale invariant, wehave sup B g ( x, /T ) | φ i | (cid:46) sup B g ( x, /T ) × [ − /T, /T ] | h | (cid:46) || h T || L ∞ ( B ) (4.1) XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 11 sup B g ( x, /T ) |∇ φ i | (cid:46) sup B g ( x, /T ) × [ − /T, /T ] |∇ h | (cid:46) T || h T || C ( B ) , (4.2)where B = B g ( x, × [ − , h is an harmonic function (∆ h = 0), for any k ≥ || h T || H k ( B ) (cid:46) k || h T || L (2 B ) (4.3)where H k is the Sobolev’s norm. Bounding the C norm by the H k norm for k sufficientlylarge depending on the dimension of M , the lemma follows by inserting (4.3) into (4.1)and (4.2), and noticing that || h T || L (2 B ) (cid:46) || φ i,x || L ( B (0 , . (cid:3) Proof of Lemma 4.2.
First, we observe that, given p ≥
2, the function x → x p/ is convexfor x ≥
0. Therefore, applying Jensen’s inequality to part (1) of Lemma 4.3, we obtain (cid:32) sup B g ( x, /T ) | φ i | (cid:33) p (cid:46) p (cid:18)(cid:90) B (0 , | φ i,x ( y ) | dy (cid:19) p/ (cid:46) p (cid:90) B (0 , | φ i,x ( y ) | p dy (4.4)and, similarly (cid:32) sup B g ( x, /T ) | T − ∇ φ i | (cid:33) p (cid:46) p (cid:90) B (0 , | φ i,x ( y ) | p dy. (4.5)We are now going to prove part (1) of Lemma 4.2. By Sogge’s bound (3.7) with p ≤ n + 1) / ( n − || φ i || L = 1, we have (cid:18)(cid:90) M | φ i ( x ) | p dx (cid:19) /p (cid:46) T n − ( − p ) =: ˜ T for all λ i ≤ T . Thus, integrating both sides of (4.4) with respect to x ∈ M and exchangingthe order of the integrals, we obtain (cid:90) M (cid:32) sup B g ( x, /T ) | φ i | (cid:33) p dx (cid:46) (cid:90) B (0 , (cid:90) M | φ i,x ( y ) | p dxdy (cid:46) ˜ T p . Therefore, by Chebyshev’s bound, for any K >
0, we haveVol g (cid:32)(cid:40) x ∈ M : sup B g ( x, /T ) | φ i | ≥ K (cid:41)(cid:33) (cid:46) K − p ˜ T p , and, taking the union bound over the O ( v ( T )) choices for i , we deduceVol g (cid:32)(cid:40) x ∈ M : max λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | φ i | ≥ K (cid:41)(cid:33) (cid:46) K − p v ( T ) · ˜ T p . (4.6)Thus, taking K = K − v ( T ) / (cid:38) K − ( ρ ( T ) T n − ) / in (4.6) and recalling that ˜ T = T n − ( − p ), we haveVol g (cid:32)(cid:40) x ∈ M : max λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | φ i | ≥ K − v ( T ) / (cid:41)(cid:33) (cid:46) K p ρ ( T ) − p +1 T ν ( n,p ) , (4.7) XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 12 where ν ( n, p ) : = − p n −
12 + n − n − (cid:16) p − (cid:17) = n − (cid:16) − p (cid:17) Hence, taking p = 2( n + 1) / ( n −
1) in (4.7), we haveVol g (cid:32)(cid:40) x ∈ M : max λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | φ i | ≥ K − v ( T ) / (cid:41)(cid:33) (cid:46) K p T − , as required. Thanks to (4.5), the proof of part (2) of Lemma 4.2 follows step by step theproof of part (1). (cid:3) Convergence to Gaussian.
In this section, we prove the following:
Lemma 4.4 (Convergence of finite dimensional distributions) . Let m be some positiveinteger and recall that B = B (0 , , moreover let F x and F µ be as in section 4. Underthe assumptions of Theorem 1.1, there exists a set A ⊂ M of volume at most O (log T /T ) such that the following holds: pick m points y , ...y m ∈ B ⊂ R n , then, uniformly for x ∈ M \ A , we have ( F x ( y ) , ..., F x ( y m )) d −→ ( F µ ( y ) , ..., F µ ( y m )) T → ∞ . and for any α = ( α , ..., α n ) , where n is the dimension of M , with | α | ≤ , we have ( D α F x ( y ) , ..., D α F x ( y m )) d −→ ( D α F µ ( y ) , ..., D α F µ ( y m )) T → ∞ . Proof of Lemma 4.4.
Let φ i,x the restriction of φ i to B g ( x, /T ) and let A be the setgiven in Lemma 4.2 applied with K = (log T ) n − n +1) = (log T ) c together with the set ofexceptional points given in Proposition 3.3. By Lemma 4.2, for all x ∈ M \ A , we havemax λ i ∈ [ T − ρ ( T ) ,T ] sup B g ( x, /T ) | φ i | (cid:46) v ( T ) / (log T ) c (4.8)and, since sup B |∇ φ i,x | (cid:46) sup B g ( x, /T ) | T − ∇ φ i | , we also havemax λ i ∈ [ T − ρ ( T ) ,T ] sup B |∇ φ i,x | (cid:46) v ( T ) / (log T ) c . (4.9)To simplify the exposition, from now on, we assume that x ∈ M \ A .In order to prove the first claim of the lemma, by the multidimensional version ofLindeberg’s Central Limit Theorem and in light of Proposition 3.3, it is sufficient toprove that for any ε > v ( T ) (cid:88) λ i E [ | a i φ i,x ( y ) | | a i φ i,x ( y ) | >εv ( T ) ] → T → ∞ (4.10)uniformly for all y ∈ B , where is the indicator function and v ( T ) = c M ρ ( T ) T n − (1 + o (1)). Thanks to Lemma 3.2, and since φ i,x are deterministic, we have1 v ( T ) (cid:88) λ i E [ | a i φ i,x ( y ) | | a i φ i,x ( y ) | >εv ( T ) ] = 1 v ( T ) (cid:88) λ i | φ i,x ( y ) | E [ | a i | | a i φ i,x ( y ) | >εv ( T ) ] ≤ sup i E [ | a i | | a i φ i,x ( y ) | >εv ( T ) ] , XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 13 thus, in order to prove (4.10), it is enough to show thatsup i E [ | a i | | a i φ i,x ( y ) | >εv ( T ) ] → T → ∞ . (4.11)Thanks to (4.8), we have | a i φ i,x ( y ) | >εv ( T ) ≤ | a i | (cid:38) ε (log T ) c , thus lim T →∞ E [ | a i | | a i φ i,x ( y ) | >εv ( T ) ] ≤ lim M →∞ lim T →∞ (cid:90) Mε (log T ) c t d P ( | a i | > t ) = 0where we have swapped limits using Fubini’s Theorem and the fact that E [ | a i | ] = 1. Thisconcludes the proof of (4.11).In order to prove the second claim of the lemma, and upon recalling the second partof Proposition 3.3, again by the multidimensional version of Lindeberg’s Central LimitTheorem, it is enough to prove that for any ε > | α | ≤ v ( T ) (cid:88) λ i E [ | a i D α φ i,x ( y ) | | a i D α φ i,x ( y ) | >εv ( T ) ] → T → ∞ (4.12)uniformly for all y ∈ B . Similarly to the above argument, (4.9) implies (4.12) if | α | = 1;for | α | = 2 we use the Helmholtz’s equation to bound the second derivatives. (cid:3) Tightness.
A sequence of probability measures v n on some topological space X is tight if for every (cid:15) >
0, there exists a compact set K = K ( (cid:15) ) ⊂ X such that ν n ( X \ K ) ≤ (cid:15), uniformly for all n ≥
0. We will need the following lemma, borrowed from [44, Lemma1], see also [7, Chapter 6 and 7], which characterise tightness in the space of continuouslytwice differentiable functions:
Lemma 4.5 (Tightness) . Let V be a compact subset of R n , let ν n be a sequence of prob-ability measures on C ( V ) , continuously twice differentiable functions on V , then ν n istight if the following conditions hold:(1) For any | α | ≤ , there exists some y ∈ V such that for every ε > there exists M > with ν n ( g ∈ C ( V ) : | D α g ( y ) | > M ) ≤ ε. (2) For any | α | ≤ and ε > , we have lim δ → lim sup n →∞ ν n (cid:32) g ∈ C ( V ) : sup | y − y (cid:48) |≤ δ | D α g ( y ) − D α g ( y (cid:48) ) | > ε (cid:33) = 0 . We wish to apply Lemma 4.5 to V = B with ν T,x being the sequence of probabilitymeasures on C ( V ) induced by the pushforward measure of F x . That is, for an open set F ⊂ C ( V ), we let ν T,x ( F ) := ( F x ) ∗ P ( A ) = P ( F x ( ω, · ) ∈ F ) . (4.13)We then have the following: Lemma 4.6.
Let V = B and let ν T,x be as in (4.13) , then, under the assumptions ofTheorem 1.1, for almost all x ∈ M the sequence ν T is tight. XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 14
Proof.
For brevity let us write ν T = ν T,x . For condition (1) of Lemma 4.5, we observethat Proposition 3.3 implies that E (cid:2) | D α F x (0) | (cid:3) (cid:46) , for | α | ≤
2. Thus, Chebyshev’s inequality give P ( | D α F x (0) | > M ) (cid:46) M , and condition (1) follows by taking M = (cid:15) − / .For condition (2) of Lemma 4.5, since F x is almost surely analytic, we havesup | y − y (cid:48) |≤ δ | D α F x ( y ) − D α F x ( y (cid:48) ) | (cid:46) sup B |∇ D α F x | δ. (4.14)Therefore it is sufficient to prove the following claim: P (sup B |∇ D α F x | > M ) (cid:46) M − . (4.15)Indeed, (4.15) together with (4.14) imply condition (2) by choosing M = (cid:15)δ − .We are now going to prove (4.15). First we observe that, for any fixed k ≥
0, Proposition3.3 gives E [ | D α F x ( y ) | ] (cid:46) y ∈ B and | α | ≤ k , where the constant implied in the (cid:46) -notation isabsolute. Now Sobolev’s inequality gives that || F x || C ( B ) (cid:46) || F x || H t ( B ) for some t sufficiently large depending on n , and the constant implied in the (cid:46) -notationis independent of T . Therefore, by (4.16), we have E [ || F x || C ( B ) ] (cid:46) E [ || F x || H t ( B ) ] = O (1) (4.17)where the constants implied in the (cid:46) -notation is independent of T . The inequality (4.17)together with Chebyshev’s inequality implies (4.15). This concludes the proof of (4.15)and thus of Lemma 4.6. (cid:3) Since v T is tight, the convergence of finite-dimensional distributions implies weak (cid:63) con-vergence. That is, combining Lemma 4.4 and Lemma 4.5, we proved the following lemma,see for example [8, Section 7]: Lemma 4.7.
Let V = B and let ν ∞ the pushforward of F µ on C ( V ) , where F µ is as insection 4. Under the assumptions of Theorem 1.1, there exists a set A ⊂ M of volumeat most O (log T /T ) such that, uniformly for all x ∈ M \ A , ν T,x weak (cid:63) converges to ν ∞ inthe space of probability measures on C ( V ) . Concluding the proof of Proposition 4.1.
To conclude the proof of Proposition4.1, we just need the following Lemma, see for example [45, Lemma 6.2], which shows that V ( · ), that is the nodal volume, is a continuous map on the appropriate space of functions: Lemma 4.8.
Let B ⊂ R n be a ball, define the set C ∗ (2 B ) = { g ∈ C (2 B ) : | g | + |∇ g | > } .Then V ( · , B ) is a continuous functional on C ∗ (2 B ) . We are now in a position to prove Proposition 4.1.
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 15
Proof of Proposition 4.1.
Let A be given by Lemma 4.7. Let B = V , by Bulinskaya’slemma, see for example [52], F µ ∈ C ∗ ( V ) almost surely. Thus, Lemma 4.7 and theContinuous Mapping Theorem imply that V ( F x ) d → V ( F µ ) T → ∞ , as required. (cid:3) Remark . If ρ ( T ) → ∞ , the conclusion of Proposition 4.1 holds without the needof removing an exceptional set and the proof is considerably simpler. Indeed, for anyeigenfunction φ i , we have [60, page 39 and page 105], || φ i || L ∞ ≤ λ n − i ||∇ φ i || L ∞ ≤ λ i || φ i || L ∞ . (4.18)Thus, if ρ ( T ) → ∞ , we have || φ i || L ∞ /v ( T ) / → T → ∞||∇ φ i,x || L ∞ /v ( T ) / → T → ∞ (4.19)where v ( T ) is as in (1.3) and φ i,x is as in section 3.3. From (4.19), we obtain Lemma 4.4without the need to remove any exceptional set.On the other hand, the extreme case ρ ( T ) = 1 includes the n -dimensional sphere, where(4.18) are known to be sharp. Thus, the exceptional set cannot be removed entirely.5. V ( F x ) and the doubling index The aim of this section is to show that we can bound V ( F x ) using the doubling index of F x on B ; in doing so, we follow the work Jerison and Lebeau [36, Section 14] which,in turn, is based on [28], see also Kukavica [26, 25, 27] for a different approach. We firstdefine the doubling index , following [32, 33] and [17]: given a (geodesic) ball B ⊂ M anda function g : 3 B → R , we define the doubling index of g on B , denoted by N ( g, B ), as N ( g, B ) = log sup B | g | sup B | g | . (5.1)It will be useful to also introduce an auxiliary function f H sometimes known as the harmonic lift of f . Following [36, Page 231] (see also [33]), we define f H : M × R → R tobe the unique solution of(∆ + ∂ t ) f H ( x, t ) = 0 f H ( x,
0) = 0 ∂ t f H ( x,
0) = f. (5.2)Note that we can write explicitly f H as f H ( x, t ) = v − / ( T ) (cid:88) λ i ∈ [ T − ρ ( T ) ,T ] a i sinh( λ i t ) λ i φ i ( x ) , (5.3)where a i and φ i are as in (1.2) and v ( T ) is as in (1.3). Given a “ball”, of radius r > B ⊂ M × R , which is defined to be B = ˜ B × I for a geodesic ball ˜ B ⊂ M of radius r > I ⊂ R of length r >
0, we denote by B + := B ∩ ( M × [0 , ∞ )) and,given r > rB + := rB ∩ ( M × [0 , ∞ )). We also define the doubling index of a function g : 3 B ⊂ M × R on B or B + analogously to (5.1). With this notation in mind, in thissection we prove the following result: XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 16
Proposition 5.1.
Let f and f H be as in (1.2) and (5.3) respectively. Then there existssome η = η ( M ) > such that the following holds: for every ball B = B ( x, r ) ⊂ M × R centred at a point lying on x ∈ M ∼ = M × { } of radius r < η/ , we have V (cid:18) f, B ∩ M (cid:19) (cid:46) N ( f H , B + ) . Our main tool to prove Proposition 5.1 is the following general result which we borrowfrom [36, Theorem 14.7] and [17, Proposition 6.7]:
Lemma 5.2.
Let B C ⊂ C n be a ball of radius , and let H be a holomorphic function on B C . If | H | L ∞ (2 B C ) ≤ e CN | H | L ∞ ( B C ) for some C > , then H n − (cid:18) { H = 0 } ∩ B ∩ R n (cid:19) (cid:46) N. To ease the exposition, we split the proof of Proposition 5.1 into a series of steps: usingthe fact that M is real analytic we will bound V ( f, · ) by the order of growth of the complexextension of f , defined in section 5.1 below. We will then show how N ( f H , · ) controls thesaid order of growth.5.1. Complexification of f . Since (
M, g ) is real analytic and compact, by the Bruhat-Whitney Theorem [54] there exists a complex manifold M C where M embeds as a totallyreal manifold. Moreover, it is possible to analytically continue any Laplace eigenfunction φ i to a homomorphic function φ C i defined on a maximal uniform Grauert tube , that isthere exists some η > φ C i is an homomorphic function on M C η := { ζ ∈ M C : √ γ ( ζ ) < η } (5.4)where √ γ ( · ) is the Grauert tube function, for details see [60, Chapter 14]. For notationalbrevity, from now on we will write M C = M C η , as the precise value of η will be unimportant,and let f C , defined on M C , be the complexification of f . The next Lemma quantify thegrowth of f C in terms of the doubling index of f H . Lemma 5.3.
Let f be as in (1.2) , f H be as in (5.2) and let f C be the complexification of f . Let B ⊂ M × R be a ball centred at a point lying on M ∼ = M × { } of radius less than η/ , where η is as in (5.4) . Suppose that, for some N > , one has || f H || L ∞ (3 B + ) (cid:46) e N || f H || L ∞ ( B + ) then || f C || L ∞ (2( B ∩ M ) C ) ≤ C (cid:48) e CN || f || L ∞ ( B ∩ M ) . for some absolute constant C > and C (cid:48) = C (cid:48) ( η ) > . To prove Lemma 5.3, we first need the following result borrowed from [36, Page 231].
Lemma 5.4.
Let B = B ( x, r ) ⊂ M × R be a ball centred at a point lying on M ∼ = M × { } of radius r > . Then, uniformly for all B , there exists some < β < such that || f H || L ∞ ( B + ) (cid:46) β,M || ∂ t f H || βL ∞ ( B ∩ M ) || f H || − βL ∞ (2 B + ) . We are now ready to prove Lemma 5.3
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 17
Proof of Lemma 5.3.
Since f H satisfies( ∂ t + ∆) f H = 0 , the hypoellipticity of the operator ∂ t + ∆, see for example [21, Lemma 7.5.1 and equation(4.4.1)], for any multi-index | α | >
0, givessup B + | D α f H | (cid:46) C | α | || f H || H ( B + ) (5.5)for some C >
1, where H is the Sobolev norm. By elliptic regularity (we can comparethe H norm of f H with the L ∞ norm of f H , see [18, Page 330]), we also have || f H || H ( B + ) (cid:46) || f H || L ∞ (3 B + ) . (5.6)Moreover, the assumption on the doubling index of f H and Lemma 5.4 give || f H || L ∞ (3 B + ) (cid:46) e N || f H || L ∞ ( B + ) (cid:46) e CN || f || L ∞ ( B ∩ M ) (5.7)for some C >
0. Therefore, putting (5.5), (5.6) and (5.7) together, we obtainsup B ∩ M ) | D α f | (cid:46) e CN + C (cid:48) | α | sup B ∩ M | f | . (5.8)Since f is real analytic, we can write f C ( z ) = (cid:88) α D α f ( x ) | α | ! z | α | thus, Lemma 5.3 follows by (5.8). (cid:3) Concluding the proof of Proposition 5.1.
Proof.
First, Lemma 5.2 gives V (cid:18) f, B ∩ M (cid:19) (cid:46) log sup B ∩ M ) C | f C | sup ( B ∩ M ) C | f C | (5.9)where f C is the complexification of f , note that here we have used the assumption that r < η/
10 where η is given as in section 5.4. Now let N = N ( f H , B + ), then Lemma 5.3imply that log sup B ∩ M ) C | f C | sup ( B ∩ M ) C | f C | (cid:46) N. (5.10)Hence, the Proposition follows by combining (5.9) and (5.10). (cid:3) Estimates for the doubling index.
In this section, we collect various estimatesfor N ( f H , · ) encountered in section 5. In light of Proposition 5.1 the said estimates willdirectly produce bounds for V ( F x ). We begin with the following estimate, see [36, page231]: Lemma 5.5.
Let f H be as in (5.2) and let B ⊂ M × R be a ball of any radius, then || f H || L ∞ (2 B + ) (cid:46) e CT || f H || L ∞ ( B + ) . for some C > . We then have the following bound on V ( F x ): Lemma 5.6.
Let F x be as in (3.3) , then V ( F x ) (cid:46) T. XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 18
Proof.
Applying Proposition 5.1 to f /T ( · ) = f ( T − · ) with B = B g ( x, × (0 ,
1) (wetacitly assume that T is sufficiently large so that 1 /T ≤ η/
10 with η as in Proposition5.1), we obtain V (cid:18) f /T , B ∩ M (cid:19) (cid:46) N ( f H /T , B ) . Since the L ∞ -norm is invariant under scaling, Lemma 5.5 gives N ( f H /T , B ) (cid:46) T . There-fore, Lemma 5.6 follows upon noticing that the definition of F x implies that V ( F x ) (cid:46) V (cid:18) f /T , B ∩ M (cid:19) . (cid:3) However, using [30, Theorem 5.3], it is possible to improve on Lemma 5.6. Indeed, wehave the following:
Theorem 5.7 (Logunov) . Let u be an harmonic function on M × R , and let O ∈ M .Then there exists constants N = N ( M, g, n, O ) and R = R ( M, g, n, O ) such that forany cube Q ⊂ B ( O, R ) the following holds: divide Q into B n subcubes, then the numberof subcubes with doubling index greater than max( N ( u, Q )2 − log B/ log log B , N ) is less than O ( B n − ) . In particular, by compactness, M can be covered by finitely many balls of radius O (1);then, applying Theorem 5.7 to each one of them with u = f H , thanks to Lemma 5.5, weobtain the following corollary: Corollary 5.8.
Let F x be as in section 4, then for almost all x ∈ M , we have V ( F x ) (cid:46) T − log T/ log log T . Anti-concentration
The aim of this section is to show that V ( F x ( · , ω )) is uniformly integrable in ω ∈ Ω,that is we prove the following Proposition:
Proposition 6.1.
Let F x be as in (3.3) , v ( T ) be as in (1.3) , and g ( t ) = t log t . Supposethat v ( T ) ≥ c M T / log T , then there exists a constant C > , independent of T such that,for almost all x ∈ M , we have E [ g ( V ( F x ))] < C. For the sake of clarity, we divide the proof of Proposition 6.1, into lemmas: we firstestimate the probability that F x attains small values, and then show how this can be usedto bound the nodal volume.6.1. Small values of f . To estimate the probability of occurrence of small of f , we needthe following lemma of Halasz, see for example [20] and [41, Lemma 6.2]. Lemma 6.2 (Halasz’ inequality) . Let X be a real-valued random variable, and let ψ ( t ) = E [exp( itX )] be its characteristic function. Then there exists some absolute constant C > such that P ( | X | ≤ ≤ C (cid:90) | t |≤ | ψ ( t ) | dt. We then prove the following Lemma:
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 19
Lemma 6.3.
Let f be as in (1.2) and v = v ( T ) = c M ρ ( T ) T n − as in (1.3) . Then,uniformly for x ∈ M , we have P ( | f ( x ) | ≤ τ ) (cid:46) τ + v ( T ) − / where the constant implied in the (cid:46) -notation is absolute.Proof. By Lemma 6.2, we have P ( | f ( x ) | /τ ≤ ≤ C (cid:90) | t |≤ | E [exp( it | f ( x ) | /τ )] | dt = Cτ (cid:90) | t |≤ /τ | E [exp( it | f ( x ) | )] | dt = Cτ (cid:90) | t |≤ /τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:89) λ i ψ i ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt (6.1)where ψ i ( t ) = E [exp( ita i φ i ( x ) /v / )] and C is as in Lemma 6.2.Now, let K > v − (cid:88) λ i | φ i ( x ) | (cid:46) , thus there are at most O ( v/K ) λ i ’s such that | φ i ( x ) | > K . We claim that, if | φ i ( x ) | ≤ K ,there exists some c > | ψ i ( t ) | ≤ e − t v − (6.2)for all t ∈ [ − cv / /K, cv / /K ]. Indeed, if | φ i ( x ) | ≤ K , then a i φ i ( x ) v − / is a randomvariable with mean zero and second moment bounded by K /v . Thus, using the fact that | exp( iy ) − (1 + iy − − y ) | ≤ min(6 − y , y ), we can write (cid:12)(cid:12)(cid:12)(cid:12) ψ i ( t ) − (cid:18) − K v t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K v t (cid:12)(cid:12)(cid:12)(cid:12) e − t v − − (cid:18) − K v t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K v t , (6.3)provided t ∈ [ − cv / /K, cv / /K ] for some sufficiently small c >
0. Therefore, (6.2)follows from (6.3), choosing c > v (1 − O ( K − )) values of i such that | φ i ( x ) | ≤ K , for all t ∈ [ − cv / /K, cv / /K ], we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:89) | φ i ( x ) |≤ K ψ i ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) exp (cid:0) − t (cid:0) − O ( K − ) (cid:1)(cid:1) . Thus, using the trivial bound | ψ i ( t ) | ≤ | φ i ( x ) | ≥ K , and picking K = 1000, for all t ∈ [ − c v / , c v / ], we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:89) i ψ i ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) exp (cid:0) − c t (cid:1) , for some absolute constant c > . Hence, if 1 /τ ≤ c v / , that is τ ≥ c − v − / , theintegral in (6.1) is bounded, thus P ( | f ( x ) | ≤ τ ) (cid:46) τ. XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 20 If τ ≤ c − v − / , we have P ( | f ( x ) | ≤ τ ) ≤ P ( | f ( x ) | ≤ c − v − / ) (cid:46) v − / . This concludes the proof of Lemma 6.3. (cid:3)
A concentration inequality for the nodal volume.
Lemma 6.3 allows us tocontrol the nodal volume of F x , as follows: Lemma 6.4.
Let F x be as in (3.3) and, let K > be some parameter. Then, for all < Q (cid:46) v / /K , we have P ( V ( F x ) > Q ) (cid:46) QK + ( QK ) e Q Proof.
Given some τ > P ( V ( F x ) > Q ) = P ( V ( F x ) > Q and | f ( x ) | < τ ) + P ( V ( F x ) > Q and | f ( x ) | ≥ τ ) . (6.4)The first term on the r.h.s. of (6.4) can be bounded, via Lemma (6.3), as P ( V ( F x ) ≥ Q and | f ( x ) | < τ ) ≤ P ( | f ( x ) | ≤ τ ) (cid:46) τ, (6.5)provided τ (cid:38) v ( T ) − / . Let us now consider the second term on the r.h.s. of (6.4), thus,we may assume that | f ( x ) | > τ . Let B = B ( x, /T ) × ( − /T, /T ), by Proposition 5.1,maintaining the same notation, we have V ( F x ) (cid:46) log || f H || L ∞ (4 B + ) || f H || L ∞ (2 B + ) ≤ log || f H || L ∞ (4 B + ) | f ( x ) | , (6.6)thus P ( V ( F x ) ≥ Q and | f ( x ) | ≥ τ ) ≤ P (cid:0) || f H || L ∞ (2 B + ) > τ e Q (cid:1) . (6.7)Now we claim the following: E [ || f H || L ∞ (4 B + ) ] = O (1) . (6.8)Indeed, writing f H /T ( · ) = f H ( T − · ) and ˜ B = B ( x, × ( − , || f H || L ∞ (4 B + ) ≤ || f H /T || L ∞ (4 ˜ B ) (cid:46) || f H /T || L (6 ˜ B ) . Therefore, using the formula (5.3) and exchanging the expectation with the sum, we seethat E [ || f H || L ∞ (4 B + ) ] (cid:46) E [ || f H /T || L (6 ˜ B ) ] (cid:46) v ( T ) − (cid:90) B (0 , (cid:88) λ i | φ i ( T − x ) | dx, thus (6.8) follows from Lemma 3.2. Using (6.8) together with Chebyshev’s inequality,(6.5) and (6.7), we obtain P ( V ( F x ) ≥ Q ) (cid:46) τ + 1 τ e Q provided that τ (cid:38) v ( T ) − / . Hence, Lemma 6.4 follows by taking τ = 1 /QK , which gives Q ≤ v ( T ) / /K . (cid:3) XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 21
Concluding the proof of Proposition 6.1.
We are finally ready to conclude theproof of Proposition 6.1:
Proof of Proposition 6.1.
By Corollary 5.8, we may assume that V ( F x ) (cid:46) T − log T/ log log t =: h ( T ) . Therefore, letting g = t log t and integrating by parts, we have E [ g ( V ( F x ))] = (cid:90) h ( T )0 g ( t ) d P ( V ( F x ) > t ) = (cid:90) h ( T )0 g (cid:48) ( t ) P ( V ( F x ) > t ) dt = (cid:90) h ( T )1 g (cid:48) ( t ) P ( V ( F x ) > t ) dt + O (1) . (6.9)Let K = K ( t ) > E [ g ( V ( F x ))] (cid:46) (cid:90) h ( T )1 g (cid:48) ( t ) 1 tK ( t ) dt + (cid:90) h ( T )1 g (cid:48) ( t ) tK ( t ) e t dt + O (1) , (6.10)provided h ( T ) ≤ v / ( T ) /K ( h ( T )). Taking K ( t ) = log( t ) , we have K ( h ( T )) (cid:38) (log T ) ,thus, using (1.3) and the assumption v ( T ) ≥ T / log T , we have h ( T ) (cid:46) T n − (log T ) ≤ v ( T ) / /K ( h ( T )) . Hence, Proposition 6.1 follow from (6.10). (cid:3) Proof of Theorem 1.1
Before concluding the proof of Theorem 1.1, we need two standard lemmas.7.1.
Preliminary lemmas.
For the reader’s convenience, we include the proofs of thefollowing two lemmas.
Lemma 7.1.
Let F x be as in (3.3) and ω n be the volume of the ball of radius in R n .Then, we have V ( f ) = 2 n Tω n (1 + o T →∞ (1)) (cid:90) M V ( F x ) dV g ( x ) . Proof.
First, we observe that we may write V (cid:18) f, B g (cid:18) x, T (cid:19)(cid:19) = (cid:90) f − (0) y ∈ B g ( x, / (2 T )) d H n − ( y ) . (7.1)Then, integrating both sides of (7.1) and using Fubini’s Theorem, we have (cid:90) M V (cid:18) f, B g (cid:18) x, T (cid:19)(cid:19) dV g ( x ) = (cid:90) f − (0) Vol g (cid:18) B g (cid:18) y, T (cid:19)(cid:19) d H n − ( y ) . (7.2)Now, we observe that, in light of the definition of F x in section 4, we have V (cid:18) f, B g (cid:18) x, T (cid:19)(cid:19) · T n − (1 + o T →∞ (1)) = V ( F x ) . Thus, the LHS of (7.2) is (cid:90) M V (cid:18) f, B g (cid:18) x, T (cid:19)(cid:19) dV g ( x ) = 1 T n − (1 + o T →∞ (1)) (cid:90) M V ( F x ) dx (7.3) XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 22
Moreover, for all y ∈ M , we also haveVol g (cid:18) B g (cid:18) y, T (cid:19)(cid:19) = Vol R n (cid:18) B (cid:18) , T (cid:19)(cid:19) (cid:0) O (cid:0) T − (cid:1)(cid:1) = ω n (2 T ) n (1 + O ( T − )) . Thus, the r.h.s. of (7.2) is (cid:90) f − (0) Vol g (cid:18) B g (cid:18) y, T (cid:19)(cid:19) d H n − ( y ) = ω n (2 T ) n (1 + O ( T − )) V ( f ) (7.4)Hence, Lemma 7.1 follows from inserting (7.3) and (7.4) into (7.2). (cid:3) Lemma 7.2.
Let F µ be as in section 4, and ω n be the volume of the unit ball in R n . Then,we have E [ V ( F µ )] = 2 − n ω n (cid:18) πn (cid:19) / Γ (cid:0) n +12 (cid:1) Γ (cid:0) n (cid:1) . Proof.
Since the support of µ , being the unit sphere, is not contained in an hyperplane, thedistribution of ( F µ , ∇ F µ ) is non-degenerate. Thus, we may apply the Kac-Rice formula[2, Theorem 6.1], to see that E [ V ( F µ )] = (cid:90) − B E [ |∇ F µ ( y ) || F µ ( y ) = 0] ϕ F µ ( y ) (0) dy, (7.5)where ϕ F µ ( y ) (0) is the density of F µ ( y ) at the point 0. Since E [ | F µ ( y ) | ] = 1, ∇ F µ and F µ are independent, and bearing in mind that F µ is stationary (that is, F µ ( y ) has the samedistribution as F µ (0)), we have E [ |∇ F µ ( y ) || F µ ( y ) = 0] ϕ F µ ( y ) (0) = E [ ∇ F µ (0)] ϕ F µ (0) (0) . (7.6)The latter can be computed explicitly, see for example [47, Proposition 4.1], to be E [ |∇ F µ (0) | ] ϕ F µ (0) (0) = (cid:18) πn (cid:19) / Γ (cid:0) n +12 (cid:1) Γ (cid:0) n (cid:1) . (7.7)Hence, Lemma 7.2 follows by inserting (7.7) into (7.5) via (7.6). (cid:3) Concluding the proof of Theorem 1.1.
Proof of Theorem 1.1.
Thanks to Lemma 7.1 and Fubini’s Theorem, we have E [ V ( f )] = 2 n Tω n (1 + o T →∞ (1))) (cid:90) M E [ V ( F x )] dV g ( x ) (7.8)Thanks to Propositions 4.1, Proposition 6.1, whose assumptions are satisfied thanks tothe assumptions of Theorem 1.1, and the Dominated Convergence Theorem, we have E [ V ( F x )] = E [ V ( F µ )](1 + o T →∞ (1)) , (7.9)uniformly for all x ∈ M outside a subset A ⊂ M of volume at most O (log T /T ). Moreover,thanks to Corollary (5.8), for almost all x ∈ M , we have V ( F x ) (cid:46) T − log T/ log log T . Thus, using (7.9), we may rewrite the integral in (7.8) as (cid:90) M E [ V ( F x )] dV g ( x ) = (cid:90) M \ A E [ V ( F x )] dV g ( x ) + (cid:90) A E [ V ( F x )] dV g ( x )= Vol( M \ A ) E [ V ( F µ )](1 + o T →∞ (1)) + o T →∞ (1)= Vol( M ) E [ V ( F µ )](1 + o T →∞ (1)) + o T →∞ (1) . (7.10) XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 23
Inserting (7.10) into (7.8) and using Lemma 7.2 we obtain E [ V ( f )] = 2 n ω n Vol( M ) E [ V ( F µ )] · ( T + o T →∞ ( T ))= Vol( M ) (cid:18) πn (cid:19) / Γ (cid:0) n +12 (cid:1) Γ (cid:0) n (cid:1) T + o T →∞ ( T ) , as required. (cid:3) Appendix A. Important case: random non-Gaussian spherical harmonics
The sequence of Laplace eigenvalues on S , the two dimensional round unit sphere,is given by { (cid:96) ( (cid:96) + 1) } , where (cid:96) > (cid:96) + 1 and the corresponding Laplace eigenfunctions are the restrictions ofhomogeneous harmonic polynomials of degree (cid:96) to S , also known as spherical harmonics.There exists a “distinguished”orthonormal base for the space of spherical harmonics ofdegree (cid:96) , see for example [60, Theorem 4.9], Y (cid:96),m ( θ, ϕ ) = (cid:18) (2 (cid:96) + 1) ( (cid:96) − m )!( (cid:96) + m )! (cid:19) / P m(cid:96) (cos( θ )) e imϕ , (A.1)where m = − (cid:96), ..., (cid:96) , ( θ, ϕ ) are polar-coordinates on S and P m(cid:96) ( x ) are the associatedLegendre polynomials. Therefore, the band-limited functions on S with spectral param-eter T = (cid:96) ( (cid:96) + 1) and width of the energy window ρ ( T ) = 1, such as f (cid:96) in (2.2), can beexpressed as a linear combinations of the functions in (A.1).We observe that, under the notation (3.3) of section 3.2, thanks to the Funk-Heckeformula and the Hilb’s asymptotics for Legendre polynomials (see e.g. [38, Claim 2.7 andClaim 2.9] and references therein), one has E [ F x,(cid:96) ( y ) F x,(cid:96) ( y (cid:48) )] = J ( | y − y (cid:48) | ) + o (cid:96) →∞ (1) , and we can also differentiate both sides any arbitrary finite number of times.In this section, we prove the following theorem: Theorem A.1.
Let f (cid:96) ( x ) = (cid:96) (cid:88) m = − (cid:96) a m Y (cid:96),m ( x ) , where the Y (cid:96),m are given in (A.1) and assume that a (cid:96),m are i.i.d. ± Bernoulli randomvariables. Then E [ V ( f (cid:96) )] = 2 √ π(cid:96) + o (cid:96) →∞ ( (cid:96) ) . The main crux of extending the proof of Theorem 1.1 to the 2d-case is the failure ofProposition 6.1 in 2d. However, in the setting of Theorem A.1, it is possible to overcomethe said obstacle by invoking the following result due to Nazarov, Nishry and Sodin [37,Corollary 2.2] on the log-integrability of random Fourier series with Bernoulli coefficients:
Lemma A.2 ( [37, Corollary 2.2]) . Let g : S × Ω → C be a random Fourier series givenby g ( ϕ ) = ∞ (cid:88) m = −∞ a m b m e ( mϕ ) , XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 24 where { a m } m ∈ Z are i.i.d. ± Bernoulli random variables defined on the probability space Ω and { b m } m ∈ Z is a sequence of (deterministic) complex-valued coefficients so that || g || L ( S × Ω) = 1 . Moreover, let p > be an integer, then there is an absolute constant C = C ( p ) > suchthat (cid:90) S × Ω | log | g || p dν ≤ C, where ν is the product measure of the Lebesgue measure on S and P ( · ) on Ω . In fact, Lemma A.2 implies the following:
Corollary A.3.
Let f (cid:96) be as in Theorem A.1, x ∈ S , and F x = F x,(cid:96) the scaled versionof f (cid:96) as in (3.3) . Then there exists a constant C > such that π (cid:90) S × Ω V ( F x ) dσ < C, where σ is the product measure of the Lebesgue measure on S and P ( · ) on Ω .Proof. The proof of Corollary A.3 is very similar to the proof of Lemma 6.4, so we omitsome details while retaining same notation. Thanks to the inequality (6.6) and the factthat ( X + Y ) (cid:46) X + Y , we have (cid:90) S × Ω V ( F x ) dσ (cid:46) (cid:90) S × Ω (cid:18) log || f H(cid:96) || L ∞ (4 B + ) || f H(cid:96) || L ∞ (2 B + ) (cid:19) dσ (cid:46) O (1) + (cid:90) S × Ω | log | f (cid:96) || dσ, (A.2)where σ is the product measure on S × Ω. Now, we wish to use Lemma A.2 to estimatethe second term on the r.h.s. of (A.2).First, we observe that, writing f (cid:96) = f (cid:96) ( ϕ, θ ), where ( ϕ, θ ) are spherical coordinates, andusing Fubini’s Theorem, we have (cid:90) S × Ω | log | f (cid:96) || dσ ≤ π (cid:90) dθ (cid:90) S × Ω | log | f (cid:96) ( ϕ, θ ) || dν, (A.3)where dν = dϕ ⊗ d P is the product measure on S × Ω. Writing f (cid:96) ( x ) = f (cid:96) ( ϕ, θ ) andbearing in mind (A.1), we have f (cid:96) ( ϕ, θ ) = (cid:96) (cid:88) m = − (cid:96) a (cid:96),m (cid:18) ( (cid:96) − m )!( (cid:96) + m )! (cid:19) / P m(cid:96) (cos( θ )) e imϕ . Therefore, fixing θ , thanks to the normalisation E [ | f (cid:96) ( ϕ, θ ) | ] = 1, we may apply LemmaA.2 with b m = ( (cid:96) − m )!( (cid:96) + m )! 1 / P m(cid:96) (cos( θ )), to see that (cid:90) S × Ω | log | f (cid:96) || dν ≤ C for some absolute constant C >
0. Hence, thanks to (A.3), the r.h.s. of (A.2) is boundedand this concludes the proof of Corollary A.3. (cid:3)
We are finally in the position to prove Theorem A.1:
XPECTED NODAL VOLUME FOR NON-GAUSSIAN RANDOM BAND-LIMITED FUNCTIONS 25
Proof of Theorem A.1.
As in the proof of Theorem 1.1, Lemma 7.1, Fubini’s Theoremand Corollary 5.6 give E [ V ( f )] = 4 π (cid:96) · (1 + o (cid:96) →∞ (1)) (cid:90) S × Ω V ( F x ) dσ, (A.4)where σ is the product measure on S × Ω. Now, Proposition 4.1 implies that V ( F x ) d −→ V ( F µ ) (cid:96) → ∞ , in the product space S × Ω. Thus, Corollary A.3 together with the Dominated Conver-gence Theorem give (cid:90) S × Ω V ( F x ) dσ −→ (cid:90) S × Ω V ( F µ ) dσ (cid:96) → ∞ . Therefore, another application of Fubini’s Theorem together with (A.4) give E [ V ( f )] = 4 π (cid:96) · (1 + o (cid:96) →∞ (1)) (cid:90) S E [ V ( F µ )] dx (A.5)Finally, the right hand side of (A.5) can be computed via Lemma 7.2 and Theorem A.1follows. (cid:3) References [1]
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Zakhar Kabluchko: Institut f¨ur Mathematische Stochastik, Westf¨alische Wilhelms-Universit¨at M¨unster, Germany
Email address : [email protected] School of Mathematical Sciences, Tel Aviv University, Israel
Email address : [email protected] Department of Mathematics, King’s College London, UK
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