Nodal area distribution for arithmetic random waves

Abstract

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on T 3 = R 3 /Z 3 (3-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian, distribution. 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. We finally discuss higher-dimensional extensions of our result.

Nodal Intersections for Arithmetic Random Waves Against a Surface

“…with eigenvalue 4π 2 m, where a µ are complex standard Gaussian random variables 1 (i.e., we have E[a µ ] = 0 and E[|a µ | 2 ] = 1), independent save for the relations a −µ = a µ (so that F (x) is real valued). Several recent works [37,30,3,8] study the fine properties of the nodal set of (1.4). Our object of investigation is the following.…”

“…We will need the following lemma, that also appears in [8]. We have, for i, j = 1, 2, 3, i = j, and 0 ≤ k ≤ l, k + l = 4,…”

Random Waves On $\mathbb T^3$: Nodal Area Variance and Lattice Point Correlations

“…In addition to their application in the proof of Theorem 1.2, they allow for the study of finer aspects of A. In the companion paper [8], it is shown by way of Theorems 1.6 and 1.7, that in the Wiener chaos expansion of A, only the fourth order chaos component is asymptotically significant: its distribution is asymptotic to the distribution of A.…”

On 3-dimensional Berry’s model

“…Canzani and Hanin (2020) studied the universality phenomenon in general Riemannian manifolds. The reader can find results on arithmetic random waves defined on the flat torus (Cammarota, 2019;Dalmao et al, 2019) and on random spherical harmonics in Cammarota and Marinucci (2019); Fantaye et al (2019); Marinucci and Rossi (2021) and references therein, see also Rossi (2019) for a survey on both subjects. The nodal sets of Berry's planar random waves, i.e.…”