We consider a wide class of families of Gaussian fields on defined by
where the ’s are independent standard normals and is the set of solutions to the equation for some fixed elliptic polynomial with integer coefficients. The case 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 : a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except the multiples of , and a wide family of polynomials in many variables.
For these three classes of polynomials, we study the -dimensional volume of the zero set of . We compute the asymptotics, as along certain well chosen subsequences of integers, of the expectation and variance of . Moreover, we prove that in the same limit, 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 . More precisely, we need to study the number of such solutions for a fixed , as well as the number of quadruples of solutions satisfying , a.k.a. -correlations, and the rate of convergence of the (rescaled) counting measure of towards a certain limiting measure on the hypersurface . 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.
On considère une large classe de familles de champs gaussiens sur définies par
où les des variables normales standard indépendantes et est l’ensemble des solutions de l’équation pour un certain polynôme fixé à coefficients entiers. Le cas 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 : une famille de formes quadratiques définies positives en deux variables, toutes les formes quadratiques définies positives en trois variables sauf les multiples de , et une large famille de polynômes en plusieurs variables.
Pour ces trois classes de polynômes, on étudie le volume en dimension de l’ensemble des zéros de . On calcule les asymptotiques, quand , le long de certaines sous-suites d’entiers bien choisies, de l’espérance et de la variance de . De plus, on montre que dans la même limite, 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 . Plus précisément, on étudie le nombre de telles solutions pour fixé, ainsi que le nombre de quadruplets de solutions vérifiant , également appelées -corrélations, et le taux de convergence de la mesure de comptage (mise à l’échelle) de vers une mesure limite sur l’hypersurface . 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.
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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.
Maffucci, Riccardo 1; Rivera, Alejandro 1
@unpublished{AIF_0__0_0_A94_0, author = {Maffucci, Riccardo and Rivera, Alejandro}, title = {On the limiting behaviour of arithmetic toral eigenfunctions}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2024}, doi = {10.5802/aif.3630}, language = {en}, note = {Online first}, }
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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