On the Number of Nodal Domains of Toral Eigenfunctions

Buckley, Jeremiah; Wigman, Igor

doi:10.1007/s00023-016-0476-7

Cited by 35 publications

(47 citation statements)

References 15 publications

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“…arguing as in Lemma 2.1, we see that conditions (3.2) imply The material of this section is contained, in various forms, in [5,6]. We present it here for the convenience of the reader and since Proposition 3.1, as stated, does not appear in the literature.…”

Section: Proof Of Theorem 13mentioning

confidence: 81%

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

Sartori

2019

The Quarterly Journal of Mathematics

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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].

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Section: Proof Of Theorem 13mentioning

confidence: 81%

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

Sartori

2019

The Quarterly Journal of Mathematics

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“…These and other geometric features have also been intensively studied for random eigenfunctions on other compact manifolds such as the torus (Arithmetic Random Waves) and the plane (Berry's Random Waves model), see e.g. [3,4,18,22,5,2,14,6,10]; [4,34,33] for fluctuations over subdomains of the torus and of the sphere, [16] for the analysis of mass equidistributions; and [31,32] for nodal intersections, to list only some of the recent contributions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Distribution of the Critical Values of Random Spherical Harmonics

2015

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We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high-energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.

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“…However, the value of ν depends on the limiting measure. This situation is further investigated in [KW15]; see also [BW15]. Before presenting the lemma and its proof, we construct a vector field whose integral curves are used in the proof.…”

Section: A Equidistribution Of Lattice Points On Spheresmentioning

confidence: 99%