Abstract. Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.
Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics.It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.
"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the twodimensional torus [RW, KKW]. In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles.Our argument has two main ingredients. An explicit derivation of the Wiener-Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.
We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4π 2 E with growing multiplicity N → ∞, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is const √ E. Our main result is that the variance of the volume normalized by √ E is bounded by O(1/ √ N ), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.
We prove Central Limit Theorems and Stein-like bounds for the asymptotic
behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our
investigation combine asymptotic analysis of higher order moments for Legendre
polynomials and, in addition, recent results on Malliavin calculus and Total
Variation bounds for Gaussian subordinated fields. We discuss application to
geometric functionals like the Defect and invariant statistics, e.g.
polyspectra of isotropic spherical random fields. Both of these have relevance
for applications, especially in an astrophysical environment.Comment: 24 page
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high-energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.
Abstract. We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d ≥ 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity N → ∞.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to 1/ √ 2π. Our main result is that the variance of Leray measure is asymptotically 1/4πN , as N → ∞, at least in dimensions d = 2 and d ≥ 5.
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