The Number of Nodal Components of Arithmetic Random Waves

We study the number of nodal components (connected components of the set of zeroes) of functions in the ensemble of arithmetic random waves, that is, random eigenfunctions of the Laplacian on the flat $d$-dimensional torus $\mathbb{T}^{d}$ ($d\ge2$). Let $f_{L}$ be a random solution to $\Delta f+4\pi^{2}L^{2}f=0$ on $\mathbb{T}^{d}$, where $L^{2}$ is a sum of $d$ squares of integers, and let $N_{L}$ be the random number of nodal components of $f_{L}$. By recent results of Nazarov and Sodin, $\mathbb{E}\left\{ N_{L}/L^{d}\right\}$ tends to a limit $\nu>0$, depending only on $d$, as $L\to\infty$ subject to a number-theoretic condition - the equidistribution on the unit sphere of the normalized lattice points on the sphere of radius $L$. This condition is guaranteed when $d\ge5$, but imposes restrictions on the sequence of $L$ values when $2\le d\le4$. We prove the exponential concentration of the random variables $N_{L}/L^{d}$ around their medians and means (unconditionally) and around their limiting mean $\nu$ (under the condition that it exists).