Central Limit Theorem for Planck‐scale Mass Distribution of Toral Laplace Eigenfunctions

Abstract

We study the fine scale L 2 -mass distribution of toral Laplace eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed in a more restrictive scenario ("Bourgain's eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.

Mass distribution for toral eigenfunctions via Bourgain’s de-randomization

“…Via the study of the variance, we also show that, even for "generic" sequences of eigenfunctions, the mass might not equidistribute at Planck scale. Moreover, we are able to give sufficient and necessary conditions for mass equidistribution which include and extend some of the results from [10,22].…”

“…Therefore, (1.2) holds for most points x ∈ T 2 . Furthermore, Wigman and Yesha [22] proved, under flatness assumptions and small variation of the coefficients (see Remark 1.6 below), that the distribution of M f (x, r) is asymptotically Gaussian with mean zero and variance c · ( √ Er) −1 , where the constant c depends on the eigenfunction f .Bourgain [2] observed that "generic" toral eigenfunctions, when averaged over T 2 , are comparable to a Gaussian random field. We apply the so called Bourgain's de-randomisation to study M f (x, r).…”

Small Scale Equidistribution for a Point Scatterer on the Torus

“…Further results are due to Han [9] (small scale equidistribution for random eigenbases on a certain class of "symmetric" manifolds), Han and Tacy [10] (random combinations of Laplace eigenfunctions on compact manifolds), Humphries [14] (small scale equidistribution for Hecke-Maass forms, with balls A = B r (x) whose centres are random. See also [7,28] for results on the torus), and de Courcy-Ireland [6] (discrepancy estimates for random spherical harmonics).…”

On the Random Wave Conjecture for Dihedral Maaß Forms

“…The asymptotic formula (1.2) is equivalent to the pointwise convergence of X g;R to 1, while (1.3) is simply the convergence in probability of X g;R to 1, a consequence of the bound Var(X g;R ) = o(1). One could ask for further refinements of these problems, such as asymptotic formulae for this variance and a central limit theorem, as studied in [WY19] for toral Laplace eigenfunctions, though we do not pursue these problems.…”