On the limiting behaviour of arithmetic toral eigenfunctions

Maffucci, Riccardo; Rivera, Alejandro

doi:10.5802/aif.3630

On the limiting behaviour of arithmetic toral eigenfunctions
[Sur le comportement limite des fonctions propres arithmétiques sur le tore]

Maffucci, Riccardo ¹ ; Rivera, Alejandro ¹

¹ EPFL, MA SB Batiment 8, Lausanne (Switzerland)

Annales de l'Institut Fourier, Online first, 59 p.

Résumé
Abstract

On considère une large classe de familles ${(F_{m})}_{m \in ℕ}$ de champs gaussiens sur $𝕋^{d} = ℝ^{d} / ℤ^{d}$ définies par

F_{m} : x ⟼ \frac{1}{\sqrt{| Λ_{m} |}} \sum_{λ \in Λ_{m}} ξ_{λ} e^{i 2 π 〈 λ, x 〉}

où les $ξ_{λ}$ des variables normales standard indépendantes et $Λ_{m}$ est l’ensemble des solutions $λ \in ℤ^{d}$ de l’équation $p (λ) = m$ pour un certain polynôme fixé $p$ à coefficients entiers. Le cas $p (x) = x_{1}^{2} + \dots + x_{d}^{2}$ revient à considérer une fonction propre aléatoire du laplacien dont la loi est parfois appelée onde arithmétique aléatoire et a été étudiée par de nombreux auteurs. À l’inverse, on considère trois classes de polynômes $p$ : une famille de formes quadratiques définies positives en deux variables, toutes les formes quadratiques définies positives en trois variables sauf les multiples de $x_{1}^{2} + x_{2}^{2} + x_{3}^{2}$ , et une large famille de polynômes en plusieurs variables.

Pour ces trois classes de polynômes, on étudie le volume $𝒱_{m}$ en dimension $d - 1$ de l’ensemble des zéros de $F_{m}$ . On calcule les asymptotiques, quand $m \to + \infty$ , le long de certaines sous-suites d’entiers bien choisies, de l’espérance et de la variance de $𝒱_{m}$ . De plus, on montre que dans la même limite, $\frac{𝒱_{m} - 𝔼 [𝒱_{m}]}{\sqrt{V a r (𝒱_{m})}}$ converge vers une loi normale standard.

Comme dans des travaux antérieurs analogues sur ce sujet pour l’onde arithmétique aléatoire, une méthode très générale réduit le problème de ces asymptotiques à l’étude de certaines propriétés arithmétiques de l’ensemble des solutions de $p (λ) = m$ . Plus précisément, on étudie le nombre de telles solutions pour $m$ fixé, ainsi que le nombre de quadruplets de solutions $(λ, μ, ν, ι)$ vérifiant $λ + μ + ν + ι = 0$ , également appelées $4$ -corrélations, et le taux de convergence de la mesure de comptage (mise à l’échelle) de $Λ_{m}$ vers une mesure limite sur l’hypersurface ${p (x) = 1}$ . Pour cela, on utilise de nombreux résultats antérieurs sur le sujet mais on établit aussi une nouvelle estimation sur les corrélations qui présente son propre intérêt.

We consider a wide class of families ${(F_{m})}_{m \in ℕ}$ of Gaussian fields on $𝕋^{d} = ℝ^{d} / ℤ^{d}$ defined by

F_{m} : x ⟼ \frac{1}{\sqrt{| Λ_{m} |}} \sum_{λ \in Λ_{m}} ξ_{λ} e^{i 2 π 〈 λ, x 〉}

where the $ξ_{λ}$ ’s are independent standard normals and $Λ_{m}$ is the set of solutions $λ \in ℤ^{d}$ to the equation $p (λ) = m$ for some fixed elliptic polynomial $p$ with integer coefficients. The case $p (x) = x_{1}^{2} + \dots + x_{d}^{2}$ amounts to considering a random Laplace eigenfunction whose law is sometimes called the arithmetic random wave and has been 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 the multiples of $x_{1}^{2} + x_{2}^{2} + x_{3}^{2}$ , and a wide family of polynomials in many variables.

For these three classes of polynomials, we study the $(d - 1)$ -dimensional volume $𝒱_{m}$ of the zero set of $F_{m}$ . We compute the asymptotics, as $m \to + \infty$ along certain well chosen subsequences of integers, of the expectation and variance of $𝒱_{m}$ . Moreover, we prove that in the same limit, $\frac{𝒱_{m} - 𝔼 [𝒱_{m}]}{\sqrt{V a r (𝒱_{m})}}$ converges to a standard normal.

As in previous analogous works on this topic for the arithmetic random wave, a very general method reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p (λ) = m$ . More precisely, we need to study the number of such solutions for a fixed $m$ , as well as the number of quadruples of solutions $(λ, μ, ν, ι)$ satisfying $λ + μ + ν + ι = 0$ , a.k.a. $4$ -correlations, and the rate of convergence of the (rescaled) counting measure of $Λ_{m}$ towards a certain limiting measure on the hypersurface ${p (x) = 1}$ . To this end, we use many previous results on this topic but also prove a new estimate on correlations which may be of independent interest.

Reçu le : 2021-11-05
Révisé le : 2023-02-03
Accepté le : 2023-05-12
Première publication : 2024-05-17

DOI : 10.5802/aif.3630

Classification : 11D72, 28C20, 35P20, 60G60, 11D45, 11P21
Keywords: Gaussian fields, Limiting theorems, Lattice points on manifolds, Equidistribution, Lattice point correlations, Kac–Rice formulas, Wiener chaos.
Mot clés : Champs gaussiens, théorèmes limite, réseaux de points sur les variétés, équidistribution, corrélations de points de réseaux, formules de Kac–Rice, chaos de Wiener.

Affiliations des auteurs :

Maffucci, Riccardo ¹ ; Rivera, Alejandro ¹

¹ EPFL, MA SB Batiment 8, Lausanne (Switzerland)

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Maffucci, Riccardo; Rivera, Alejandro. On the limiting behaviour of arithmetic toral eigenfunctions. Annales de l'Institut Fourier, Online first, 59 p.

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