On the limiting behaviour of arithmetic toral eigenfunctions

TL;DR

This paper investigates the asymptotic behavior of the zero sets of Gaussian eigenfunctions on tori derived from solutions to polynomial equations, establishing normal convergence and analyzing arithmetic properties of solution sets.

Contribution

It extends previous work by analyzing a broader class of polynomials and provides new estimates on solution correlations for eigenfunction zero sets.

Findings

Asymptotic formulas for expectation and variance of zero set volume

Normal distribution convergence of normalized zero set volume

New estimates on solution correlations and solution counting measures

Abstract

We consider a wide class of families $(F_m)_{m\in\mathbb{N}}$ of Gaussian fields on $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ defined by \[F_m:x\mapsto \frac{1}{\sqrt{|\Lambda_m|}}\sum_{\lambda\in\Lambda_m}\zeta_\lambda e^{2\pi i\langle \lambda,x\rangle}\] where the $\zeta_\lambda$'s are independent std. normals and $\Lambda_m$ is the set of solutions $\lambda\in\mathbb{Z}^d$ to $p(\lambda)=m$ for a fixed elliptic polynomial $p$ with integer coefficients. The case $p(x)=x_1^2+\dots+x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $\textit{arithmetic random wave}$, studied in the past by many authors. In contrast, we consider three classes of polynomials $p$: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except multiples of $x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables. For these classes of polynomials, we study the $(d-1)$-dimensional volume $\mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as $m\to+\infty$ along certain sequences of integers, of the expectation and variance of $\mathcal{V}_m$. Moreover, we prove that in the same limit, $\frac{\mathcal{V}_m-\mathbb{E}[\mathcal{V}_m]}{\sqrt{\text{Var}(\mathcal{V}_m)}}$ converges to a std. normal. As in previous works, one reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p(\lambda)=m$. We need to study the number of such solutions for fixed $m$, the number of quadruples of solutions $(\lambda,\mu,\nu,\iota)$ satisfying $\lambda+\mu+\nu+\iota=0$, ($4$-correlations), and the rate of convergence of the counting measure of $\Lambda_m$ towards a certain limiting measure on the hypersurface $\{p(x)=1\}$. To this end, we use prior results on this topic but also prove a new estimate on correlations, of independent interest.