Nodal intersections for random eigenfunctions on the torus

Rudnick, Zeév; Wigman, Igor

doi:10.1353/ajm.2016.0048

Cited by 37 publications

(69 citation statements)

References 31 publications

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“…A computation shows that when Σ is a sphere or a hemisphere, the lower bound in (1.15) is achieved, hence the leading term in (1.14) vanishes: in this case the variance is of lower order than m/N (see section 7 for details). As in the problem of nodal intersections against a curve on T 2 [36], the theoretical maximum of the variance leading term is achieved in the case of intersection with a 3 Lattice points on circles equidistribute [19,20] for a density one sequence of energies. To the other extreme, Cilleruelo proved that there exist sequences s.t.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…manifold of identically zero curvature (straight lines in dimension 2, planes in dimension 3). As the case of Σ contained in a plane is excluded by the assumptions of Theorem 1.3, the upper bound of A 2 · π 2 /30 for the leading coefficient in (1.14) is a supremum rather than a maximum, as in [36] (see section 7).…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…[15,38] as m → ∞. Rudnick-Wigman [36] and subsequently Rossi-Wigman [34] and the author [31] investigated Z…”

Section: )mentioning

confidence: 99%

“…Throughout we apply many ideas of [35,26,36,37]. The arithmetic random wave F (d) : T d → R (1.6) is a random field.…”

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

“…The arithmetic random wave F (d) : T d → R (1.6) is a random field. The number of nodal intersections Z (d) (1.7) against a curve are the zeros of a process, which is the restriction of the random wave F to the curve [36,34,37]. For a smooth process p : T → R defined on an appropriate parameter set T ⊂ R, moments of the number of zeros may be computed, under certain assumptions, via Kac-Rice formulas [2,13].…”

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

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Nodal Intersections for Arithmetic Random Waves Against a Surface

Maffucci¹

2019

Ann. Henri Poincaré

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Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus ('3d arithmetic random waves'), we investigate the 1-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the 'nodal intersection length'). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry.Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of nonvanishing Gauss-Kronecker curvature. The problem is closely related to the theory of lattice points on spheres: by the equidistribution of the lattice points, the variance asymptotic depends only on the geometry of the surface.

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Section: Statement Of Main Resultsmentioning

confidence: 99%

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…[15,38] as m → ∞. Rudnick-Wigman [36] and subsequently Rossi-Wigman [34] and the author [31] investigated Z…”

Section: )mentioning

confidence: 99%

“…Throughout we apply many ideas of [35,26,36,37]. The arithmetic random wave F (d) : T d → R (1.6) is a random field.…”

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

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Nodal Intersections for Arithmetic Random Waves Against a Surface

Maffucci¹

2019

Ann. Henri Poincaré

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A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

et al. 2019

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A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.

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Nodal Statistics of Planar Random Waves

Nourdin

Peccati

Rossi

2019

Commun. Math. Phys.

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We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue E > 0, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the highenergy limit (E → ∞). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat 2-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by , in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).

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Nodal intersections for random eigenfunctions on the torus

Cited by 37 publications

References 31 publications

Nodal Intersections for Arithmetic Random Waves Against a Surface

Nodal Intersections for Arithmetic Random Waves Against a Surface

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

Nodal Statistics of Planar Random Waves

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