On the variance of the nodal volume of arithmetic random waves

TL;DR

This paper investigates the variance of the nodal set volume of arithmetic random waves on high-dimensional tori, providing new upper bounds that improve understanding of the variance's asymptotic behavior in dimensions four and higher.

Contribution

It establishes an upper bound on the variance of the nodal volume for dimensions four and above, extending previous results and leveraging recent advances in decoupling theory.

Findings

Proves an upper bound of O(E/ N^{1+α(d)-ε}) for the variance in dimensions d≥4.

Shows the bound is nearly optimal for d≥5 due to recent decoupling results.

Extends variance estimates beyond the previously known cases for d=2 and d=3.

Abstract

Rudnick and Wigman (Ann. Henri Poincar\'{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/\mathcal{N})$, as $E\to\infty$, where $E$ is the energy and $\mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})$, for any $\epsilon>0$ and $d\geq 4$, where $\alpha(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $\epsilon$) when $d\geq 5$ due to the proof of the $\ell^{2}$-decoupling conjecture by Bourgain and Demeter.