Nodal deficiency of random spherical harmonics in presence of boundary

The nodal line of a smooth function $f:M→R$⁠, defined on a smooth compact surface $M$⁠, with or without a boundary, is its zero set f⁻¹(0). If f is non-singular, i.e., f has no critical zeros, then its nodal line is a smooth curve with no self-intersections. An important descriptor of f is its nodal length, i.e., the length of f⁻¹(0), receiving much attention in the last couple of decades, in particular, concerning the nodal length of the eigenfunctions of Laplacian Δ on $M$⁠, in the high-energy limit.

Let $(ϕj,λj)j≥1$ be the Laplace eigenfunctions on $M$⁠, with energies λ_j in increasing order counted with multiplicity, i.e.,

endowed with the Dirichlet boundary conditions $ϕ|∂M≡0$ in the presence of nontrivial boundary. In this context, Yau’s conjecture asserts that the nodal length $L(ϕj)$ of ϕ_j is commensurable with $λj$⁠, in the sense that

with some constants $CM>cM>0$⁠. Yau’s conjecture was resolved for $M$ analytic,^9,10,14 and, more recently, a lower bound²² and a polynomial upper bound^21,23 were asserted in full generality (i.e., for $M$ smooth).

In his highly influential work,⁵ Berry proposed to compare the high-energy Laplace eigenfunctions on generic chaotic surfaces and their nodal lines to random monochromatic waves and their nodal lines, respectively. The random monochromatic waves (also called Berry’s “random wave model” or RWM) are a centered isotropic Gaussian random field $u:R2→R$ prescribed uniquely by the covariance function

with $x,y∈R2$ and J₀(·) being the Bessel J function.

Let

be the zero density, also called the “first intensity” function of u, with ϕ_u(x) being the probability density function of the random variable u(x). In this isotropic case, it is easy to directly evaluate

and then appeal to the Kac–Rice formula, valid under the easily verified non-degeneracy conditions on the random field u, to evaluate the expected nodal length $L(u;R)$ of u(·) restricted to a radius-R disk $B(R)⊆R2$ to be precisely

Berry⁶ found that as R → ∞, the variance $(L(u;R))$ satisfies the asymptotic law

much smaller than the a priori heuristic prediction $Var(L(u;R))≈R3$ made based on the natural scaling of the problem, due to what is now known as “Berry’s cancellation”³⁴ of the leading non-oscillatory term of the two-point correlation function (also known as the “second zero intensity”).

Furthermore, in the same work,⁶ Berry studied the effect induced on the nodal length of eigenfunctions, satisfying the Dirichlet condition on a nontrivial boundary, both in its vicinity and far away from it. With the (infinite) horizontal axis ${(x1,x2):x2=0}⊆R2$ serving as a model for the boundary, he introduced a Gaussian random field $v(x1,x2):R×R>0→R$ of boundary-adapted (non-stationary) monochromatic random waves, forced to vanish at x₂ = 0. Formally, v(x₁, x₂) is the limit, as J → ∞, of the superposition

of J plane waves of wavenumber 1 forced to vanish at x₂ = 0. Alternatively, v is the centered Gaussian random field prescribed by the covariance function

where x = (x₁, x₂), y = (y₁, y₂), and $ỹ=(y1,−y2)$ is the mirror symmetry of y; the law of v is invariant with respect to horizontal shifts,

$a∈R$⁠, but not the vertical shifts.

By comparing (1.2) to (1.7), we observe that far away from the boundary (i.e., x₂, y₂ → ∞), r_v(x, y) ≈ J₀(‖x − y‖) so that in that range, the (covariance of) boundary-adapted waves converge to the (covariance of) isotropic ones (1.2), although the decay of the error term in this approximation is slow and of oscillatory nature. Intuitively, it means that, at infinity, the boundary has a small impact on the random waves, although it takes its toll on the nodal bias, as was demonstrated by Berry, as follows.

Let $K1v(x)=K1v(x2)$ be the zero density of v, defined analogously to (1.3), depending on the height x₂ only, independent of x₁ by the inherent invariance (1.8). Berry showed that as x₂ → 0,

and attributed this “nodal deficiency” $12π<122$⁠, relative to (1.4), to the a.s. orthogonality of the nodal lines touching the boundary (Ref. 13, Theorem 2.5). Though a significant proportion of the details of the computation were omitted, we validated Berry's assertions for ourselves.

Furthermore, as x₂ → ∞,

with some prescribed error term E(·). Here E(x₂) is of order $1|x2|$⁠, so not smaller by magnitude than $132πx2$⁠, but of oscillatory nature, and will not contribute to the Kac-Rice integral along expanding domains, as neither the term $cos(2x2−π/4)πx2$⁠. In this situation, a natural choice for expanding domains is the rectangles $DR≔[−1,1]×[0,R]$⁠, R → ∞ (say). As an application of the Kac–Rice formula (1.5), in this case, it easily follows that

i.e., a logarithmic “nodal deficiency” relative to (1.5), impacted by the boundary infinitely many wavelengths away from it. The logarithmic fluctuations (1.6) in the isotropic case u, possibly also holding for v, give rise to a hope to be able to detect the said, also logarithmic, negative boundary impact (1.11) via a single sample of the nodal length or, at least, very few ones.

The (unit) sphere $M=S2$ is one of but few surfaces, where the solutions to the Helmholtz equation (1.1) admit an explicit solution. For a number $ℓ∈Z≥0$⁠, the space of solutions of (1.1) with λ = ℓ(ℓ + 1) is the (2ℓ + 1)-dimensional space of degree-ℓ spherical harmonics, and conversely, all solutions to (1.1) are spherical harmonics of some degree ℓ ≥ 0. Given ℓ ≥ 0, let $Eℓ≔{ηℓ,1,…ηℓ,2ℓ+1}$ be any L²-orthonormal basis of the space of spherical harmonics of degree ℓ. The random field

with a_k being i.i.d. standard Gaussian random variables, is the degree-ℓ random spherical harmonics.

The law of $Tℓ̃$ is invariant with respect to the chosen orthonormal basis $Eℓ$⁠, uniquely defined via the covariance function

with P_ℓ(·) being the Legendre polynomial of degree ℓ, and d(·, ·) is the spherical distance between $x,y∈S2$⁠. The random fields ${Tℓ̃}$ are the Fourier components in the L²-expansion of every isotropic random field,²⁴ of interest, for instance, in cosmology and the study of Cosmic Microwave Background (CMB) radiation.

Let $L(Tℓ̃)$ be the total nodal length of $Tℓ̃$⁠, of high interest for various pure and applied disciplines, including the above. Berard⁴ evaluated the expected nodal length to be precisely

and as ℓ → ∞, its variance is asymptotic³⁴ to

in accordance with Berry’s result (1.6), save for the scaling, and the invariance of the nodal lines with respect to the symmetry x ↦ −x of the sphere, resulting in a doubled leading constant in (1.15) relative to (1.6) suitably scaled. A more recent proof²⁵ of the central limit theorem for $L(Tℓ̃)$⁠, asserting the asymptotic Gaussianity of

is sufficiently robust to also yield the central limit theorem, as R → ∞ for the nodal length $L(u;R)$ of Berry’s random waves, as was recently demonstrated,³¹ also claimed by Ref. 27.

Our principal results concern the hemisphere $H2⊆S2$⁠, endowed with the Dirichlet boundary conditions along the equator. We will widely use the spherical coordinates

with the equator identified with ${θ=π/2}⊆H2$⁠. Here, all the Laplace eigenfunctions are necessarily spherical harmonics restricted to $H2$⁠, subject to some extra properties. Recall that a concrete (complex-valued) orthonormal basis of degree ℓ is the Laplace spherical harmonics ${Yℓ,m}m=−ℓℓ$⁠, given in the spherical coordinates by

for m ≥ 0 and $Yℓ,m(θ,ϕ)=(−1)mYℓ,−m*(θ,ϕ)$ for m < 0, with $Pℓm(⋅)$ being the associated Legendre polynomials of degree ℓ on order m. For ℓ ≥ 0 and |m| ≤ ℓ, the spherical harmonic Y_ℓ,m obeys the Dirichlet boundary condition on the equator if and only if m ≢ ℓ mod 2, spanning a subspace of dimension ℓ inside the (2ℓ + 1)-dimensional space of spherical harmonics of degree ℓ (Ref. 17, Example 4). [Its (ℓ + 1)-dimensional orthogonal complement is the subspace, satisfying the Neumann boundary condition.] Conversely, every Laplace eigenfunction on $H2$ is necessarily a spherical harmonic of some degree ℓ ≥ 0 that is a linear combination of Y_ℓ,m with m ≢ ℓ mod 2.

The principal results of this paper concern the following model of boundary-adapted random spherical harmonics:

where a_ℓ,m are the standard (complex-valued) Gaussian random variables subject to the constraint $aℓ,−m=aℓ,m̄$ so that T_ℓ(·) is real-valued. Our immediate concern is for the law of T_ℓ, which, as for any centered Gaussian random field, is uniquely determined by its covariance function, claimed by the following proposition.

It is evident, either from the definition or the covariance, that the law of T_ℓ is invariant with respect to rotations of $H2$ around the axis orthogonal to the equator, that is, in the spherical coordinates,

ϕ ∈ [0, 2π). The boundary impact of (1.17) relative to (1.13) is in perfect harmony with the boundary impact of the covariance (1.7) of Berry’s boundary-adapted model relative to the isotropic case (1.2), except that the mirror symmetry $y↦ỹ$ relative to the x axis in the Euclidean situation is substituted by mirror symmetry $y↦ȳ$ relative to the equator for the spherical geometry. These generalize to two dimensions the boundary impact on the ensemble of stationary random trigonometric polynomials on the circle,^16,32 resulting in the ensemble of non-stationary random trigonometric polynomials vanishing at the end points.^1,15

Let

be the zero density of T_ℓ, which, unlike the rotation invariant spherical harmonics (1.12), genuinely depends on $x∈H2$⁠. More precisely, by the said invariance with respect to (1.18), the zero density K_1,ℓ(x) depends on the polar angle θ only. We rescale by introducing the variable

and, with a slight abuse of notation, write

Our principal result deals with the asymptotics of K_1,ℓ(·), in two different regimes, in line with (1.9) and (1.10), respectively.

Clearly, the statement (1.22) is asymptotic for ψ small only, otherwise yielding the mere bound K_1,ℓ(ψ) = O(ℓ), which is easy. As a corollary to Theorem 1.2, one may evaluate the asymptotic law of the total expected nodal length of T_ℓ and detect the negative logarithmic bias relative to (1.14), in full accordance with Berry’s result (1.11).

Another surface admitting explicit solution to the Helmholtz equation (1.1) is the standard torus $T2=R2/Z2$⁠. Here, the Laplace eigenfunctions with eigenvalue 4π²n all correspond to an integer n expressible as a sum of two squares and are given by a sum

over all lattice points $μ=(μ1,μ2)∈Z2$ lying on the radius-$n$ centered circle, where n is a sum of two squares with e(y)≔e^2πiy, ⟨μ, x⟩ = μ₁x₁ + μ₂x₂, and $x=(x1,x2)∈T2$⁠. Following Oravecz, Rudnick, and Wigman,²⁸ one endows the eigenspace of {f_n} with a Gaussian probability measure with the coefficient a_μ standard (complex-valued) i.i.d. Gaussian, save for $a−μ=aμ̄$⁠, resulting in the ensemble of “arithmetic random waves.”

The expected nodal length of f_n was computed²⁹ to be

and the useful upper bound

was also asserted, with r₂(n) being the number of lattice points lying on the radius-$n$ circle or, equivalently, the dimension of the eigenspace {f_n} as in (2.1). A precise asymptotic law for $Var(L(fn))$ was subsequently established,¹⁹ shown to fluctuate, depending on the angular distribution of the lattice points. A non-central non-universal limit theorem was asserted²⁶ also depending on the angular distribution of the lattice points.

An instrumental key input to both the said asymptotic variance and the limit law was Bourgain’s first nontrivial upper bound (Ref. 19, Theorem 2.2) of $or2(n)→∞r2(n)4$ for the number of length-six spectral correlations, i.e., six-tuples of lattice points {μ: ‖μ‖² = n} summing up to 0. Bourgain’s bound was subsequently improved and generalized to higher order correlations,⁷ in various degrees of generality, conditionally or unconditionally. These results are still actively used within the subsequent and ongoing research, in particular,⁸ and its followers.

It makes sense to compare the torus to the square with Dirichlet boundary and test what kind of impact it would have relative to (2.2) on the expected nodal length, as the “boundary-adapted arithmetic random waves,” which were addressed in Ref. 11. It was concluded, building on the work of Bourgain and Bombieri,⁷ and by appealing to a different notion of spectral correlation, namely, the spectral semi-correlations, that, even at the level of expectation, the total nodal bias is fluctuating from nodal deficiency (negative bias) to nodal surplus (positive bias) depending on the angular distribution of the lattice points and its interaction with the direction of the square boundary, at least, for generic energy levels. A similar experiment conducted by Gnutzmann and Lois for cuboids of arbitrary dimensions, averaging for eigenfunctions admitting separation of variables belonging to different eigenspaces, revealed consistency with Berry’s nodal deficiency ansatz stemming from (1.11).

It would be useful to test whether different Gaussian random fields on the square would result in different limiting nodal biases around the boundary corresponding to (1.22), which is likely to bring in a different notion of spectral correlation, not unlikely “quasi-semi-correlation.”^3,18 Another question of interest is “de-randomize” any of these results, i.e., infer the corresponding results on deterministic eigenfunctions following the work of Bourgain.⁸ We leave all these to be addressed elsewhere.

In the analysis of K_1,ℓ(x), we naturally encounter the distribution of T_ℓ(x), determined by

and the distribution of ∇T_ℓ(x) conditioned on T_ℓ(x) = 0, determined by its 2 × 2 covariance matrix

Let x correspond to the spherical coordinates (θ, ϕ). An explicit computation shows that the covariance matrix Ω_ℓ(x) depends only on θ, and below, we will often abuse a notation to write Ω_ℓ(θ) instead, and also, when convenient, Ω_ℓ(ψ) with ψ as in (1.20). A direct computation shows the following:

In Secs. III A and III B, we prove Lemma 3.1, that is, we evaluate the 2 × 2 covariance matrix of ∇T_ℓ(x) conditioned upon T_ℓ(x) = 0. First, in Sec. III A, we evaluate the unconditional 3 × 3 covariance matrix Σ_ℓ(x) of (T_ℓ(x), ∇T_ℓ(x)), and then, in Sec. III B, we apply the standard procedure for conditioning multivariate Gaussian random variables.

The covariance matrix of

could be expressed as

where

The 1 × 2 matrix B_ℓ(x) is

where B_ℓ(x) depends only on θ, and by an abuse of notation, we write

The entries of the 2 × 2 matrix C_ℓ(x) are

where again recalling that x = (θ, ϕ), we write

The conditional covariance matrix of the Gaussian vector (∇T_ℓ(x)|T_ℓ(x) = 0) is given by the following standard Gaussian transition formula:

Again taking x = (θ, ϕ) and observing that

and

we have

which is the statement of Lemma 3.1.

The asymptotic analysis (1.21) is in two steps. First, we evaluate the variance Var(T_ℓ(x)) and each entry in S_ℓ(x) using the high degree asymptotics of the Legendre polynomials and its derivatives (Hilb’s asymptotics). In the second step, performed within Proposition 4.3, we exploit the analyticity of the Gaussian expectation (1.19) as a function of the entries of the corresponding non-singular covariance matrix, to Taylor expand K_1,ℓ(x) where both Var(T_ℓ(x)) − 1 and the entries of S_ℓ(x) are assumed to be small.

For a Proof of Lemmas 4.1 and 4.2, we refer to Ref. 30, Theorem 8.21.6 and Ref. 20, Sec. 5.11, respectively.

Recall the scaled variable ψ related to θ via (1.20) so that an application of Lemmas 4.1 and 4.2 yields that for ℓ ≥ 1 and C < ψ < ℓπ,

Observing that

we write

A repeated application of Lemmas 4.1 and 4.2 also yields an asymptotic estimate for the first couple of derivatives of the Legendre polynomials (Ref. 12, Lemma 9.3),

and

Since we have that

we have

and observing that

we obtain

The estimates in (4.1), (4.2), and (4.3) imply that for ℓ ≥ 1 and uniformly for C < ψ < ℓπ, with C > 0, we have

With the same abuse of notation as above, we write S_ℓ(ψ) ≔ S_ℓ(x) as in Lemma 3.1 and in analogous manner for its individual entries S_ij;ℓ(ψ) ≔ S_11;ℓ(x). We have

The next proposition prescribes a precise asymptotic expression for the density function K_1,ℓ(·) via a Taylor expansion of the relevant Gaussian expectation as a function of the associated covariance matrix entries.