Nodal area distribution for arithmetic random waves

Cammarota, Valentina

doi:10.1090/tran/7779

Nodal area distribution for arithmetic random waves
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Trans. Amer. Math. Soc. 372 (2019), 3539-3564 Request permission

Abstract:

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb {T}^3= \mathbb {R}^3/ \mathbb {Z}^3$ (three-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 the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp. 926–960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.

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