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

Abstract

We study the mass distribution of Laplacian eigenfunctions at Planck scale for the standard flat torus T 2 = R 2 /Z 2 . By averaging over the ball centre, we use Bourgain's de-randomisation to compare the mass distribution of toral eigenfunctions to the mass distribution of random waves in growing balls around the origin. We then classify all possible limiting distributions and their variances. Moreover, we show that, even in the "generic" case, the mass might not equidistribute at Planck scale. Finally, we give necessary and sufficient conditions so that the mass of "generic" eigenfunctions equidistributes at Planck scale in almost all balls.We are interested in the case when M is the flat torus T 2 = R 2 /Z 2 . Although the geodesic flow on T 2 is completely integrable, Lester and Rudnick [19] proved that (1.2) holds for a density one subsequence of eigenfunctions at scale r > E −1/2+o(1) . On the other hand, they also proved that there exist eigenfunctions for which (1.2) fails at the point x = 0 at Planck scale. This naturally raises the question of whether the failure of (1.2) is only limited to a small set of ball centres. To make this precise, Granville and Wigman [10] introduced the (pseudo-)random variablewhere x is drawn uniformly at random from T 2 and showed, under some additional assumptions on f (see Remark 1.6 below), that the variance of M f (x, r) tends to zero at Planck scale. 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). This allows us to find its limit distribution and variance for a wider class of eigenfunctions than [10,22]. 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].

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

“…the position and mass distribution of monochromatic isotropic waves. Our intuition regarding the possibility of carrying on the explained "de-randomisation" argument for establishing results of similar nature to the presented results was recently validated by Sartori [27].…”

The defect of toral Laplace eigenfunctions and arithmetic random waves

“…for 'most' of the ball centres on the torus), and recall that, under certain flatness conditions on f n (certainly satisfied by all f n ∈ B n ) and arithmetic conditions on n (in the spirit of the ones given in section 2.1.2 above), f n (•) exhibits [7,8] Gaussian spatial value distribution when averaged over the whole torus. Using these 'derandomisation' techniques we will be able to prove the result to follow immediately; unlike the results of [7,8] (and [30]), this is a second-order result, i.e. concerning variance (as opposed to a first order one concerning expectation).…”

The defect of toral Laplace eigenfunctions and Arithmetic Random Waves

“…for "most" of the ball centres on the torus), and recall that, under certain flatness conditions on f n (certainly satisfied by all f n ∈ B n ) and arithmetic conditions on n (in the spirit of the ones given in §2.1.2 above), f n (•) exhibits [7,8] Gaussian spatial value distribution when averaged over the whole torus. Using these "de-randomisation" techniques we will be able to prove the result to follow immediately; unlike the results of [7,8] (and [29]), this is a secondorder result (as opposed to a first order one). Moreover, since, unlike [7,8], the Gaussian input for Theorem 2.6 is not inherently contained within its statement, it seems that a more direct approach might be possible for proving Theorem 2.6.…”