The Number of Nodal Components of Arithmetic Random Waves

TL;DR

This paper investigates the asymptotic behavior of the number of nodal components of arithmetic random waves on high-dimensional tori, proving concentration results and analyzing the influence of number-theoretic conditions.

Contribution

It establishes exponential concentration of the normalized nodal component count around its mean and median, and analyzes the effect of number-theoretic conditions on the limit behavior.

Findings

Expected nodal components scale with $L^d$

Concentration around the mean and median is exponential

Limit behavior depends on number-theoretic equidistribution

Abstract

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).