Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Abstract

"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.

A reduction principle for the critical values of random spherical harmonics

“…For a correct implementation of the Kac-Rice formula, our first step is to review in Section 2 some background differential geometry material on the gradient and Hessian fields, and to compute their covariances; the properties of the resulting covariance matrices are then established in Section 3, where it is shown in particular that the covariance function for the gradient vector of random eigenfunctions evaluated at any two arbitrary points on the sphere is non-singular. These results are then used in Section 4 to prove the validity (in the L 2 (Ω) sense) of the expansion for the Kac-Rice formula into Wiener chaoses, a technique exploited in other recent papers on geometric functionals of Gaussian eigenfunctions, for instance also in [27], [24], [23], [14], [32], [4], [10], [25]. Finally, in Section 5 the expansion is analytically computed and the simple dominating term is derived.…”

Nodal area distribution for arithmetic random waves

“…We remark that in dimension d = 2 |X n (4)| = 0, for all n ∈ S, which may be seen by noting that two circles intersect in at most two points (Zygmund's trick), so the asymptotic behaviour of the nodal length studied in [21] comes form a precise analysis of the asymptotic behavior of the first three terms in (2.1):…”

“…Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to [21], the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results in [1] that establish an upper bound for the number of non-degenerate correlations of lattice points on the sphere.…”

Random nodal lengths and Wiener chaos

“…A careful inspection of (4.23) reveals that the steps of the proof of (4.24) are independent of the underlying manifold, in particular the same result holds for the toral case (i.e. the second chaotic component of the nodal length for arithmetic random waves vanishes [Ros15,MPRW16]). More generally, it is natural to guess that the same cancellation phenomenon appears for the nodal volume on so-called isotropic manifolds (compact two-point homogeneous spaces [BM16]) of any dimension (like the hyperspheres [MR15]), and multidimensional tori (see also [Cam17]).…”