| 1. |
- Cammarota, Valentina, et al.
(författare)
-
Boundary Effect on the Nodal Length for Arithmetic Random Waves, and Spectral Semi-correlations
- 2020
-
Ingår i: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 376:2, s. 1261-1310
-
Tidskriftsartikel (refereegranskat)abstract
- We test M. Berry’s ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions (“boundary-adapted arithmetic random waves”). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac–Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of “bad” subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the “spectral semi-correlations”, a concept joining the likes of “spectral correlations” and “spectral quasi-correlations” in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.
|
|
| 2. |
- Chatzakos, Dimitrios, et al.
(författare)
-
On the distribution of lattice points on hyperbolic circles
- 2021
-
Ingår i: Algebra & Number Theory. - : Mathematical Sciences Publishers. - 1937-0652 .- 1944-7833. ; 15:9, s. 2357-2380
-
Tidskriftsartikel (refereegranskat)abstract
- We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane H. The angles of lattice points arising from the orbit of the modular group PSL2(Z), and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of Z2-lattice points (with certain parity conditions) lying on circles in R2, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry; on very thin subsequences they are not invariant under rotation by π/2, unlike in the Euclidean setting where all measures have this invariance property.
|
|
| 3. |
- Granville, A., et al.
(författare)
-
The distribution of the zeros of random trigonometric polynomials
- 2011
-
Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 133:2, s. 295-357
-
Tidskriftsartikel (refereegranskat)abstract
- We study the asymptotic distribution of the number Z N of zeros of random trigonometric polynomials of degree N as N →∞. It is known that as N grows to infinity, the expected number of the zeros is asymptotic to N. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be cN for some c > 0. We prove that converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.
|
|
| 4. |
- Krishnapur, Manjunath, et al.
(författare)
-
Nodal length fluctuations for arithmetic random waves
- 2013
-
Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 177:2, s. 699-737
-
Tidskriftsartikel (refereegranskat)abstract
- Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.
|
|
| 5. |
- Kurlberg, Pär, et al.
(författare)
-
Gaussian Point Count Statistics for Families of Curves Over a Fixed Finite Field
- 2011
-
Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :10, s. 2217-2229
-
Tidskriftsartikel (refereegranskat)abstract
- We produce a collection of families of curves, whose point count statistics over becomes Gaussian for p fixed. In particular, the average number of points on curves in these families tends to infinity.
|
|
| 6. |
- Kurlberg, Pär, et al.
(författare)
-
Non-universality of the Nazarov-Sodin constant
- 2015
-
Ingår i: Comptes rendus. Mathematique. - : Elsevier BV. - 1631-073X .- 1778-3569. ; 353:2, s. 101-104
-
Tidskriftsartikel (refereegranskat)abstract
- We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.
|
|
| 7. |
- Kurlberg, Pär, et al.
(författare)
-
The defect of toral Laplace eigenfunctions and arithmetic random waves
- 2021
-
Ingår i: Nonlinearity. - : IOP PUBLISHING LTD. - 0951-7715 .- 1361-6544. ; 34:9, s. 6651-6684
-
Tidskriftsartikel (refereegranskat)abstract
- We study the defect (or 'signed area') distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario ('arithmetic random waves'). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for 'most' energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct 'esoteric' eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt's subspace theorem.
|
|
| 8. |
- Kurlberg, Pär, et al.
(författare)
-
Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves
- 2018
-
Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 330, s. 516-552
-
Tidskriftsartikel (refereegranskat)abstract
- This is a manuscript containing the full proofs of results announced in [10], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions. (C) 2018 Elsevier Inc. All rights reserved.
|
|
| 9. |
- Marinucci, Domenico, et al.
(författare)
-
On the area of excursion sets of spherical Gaussian eigenfunctions
- 2011
-
Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 52:9, s. 093301-
-
Tidskriftsartikel (refereegranskat)abstract
- The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation arising from physics and cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a uniform central limit theorem for the empirical measure, i.e., the proportion of spherical surface, where spherical Gaussian eigenfunctions lie below a level z. Our proofs borrow some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate. (C) 2011 American Institute of Physics. [doi:10.1063/1.3624746]
|
|
| 10. |
- Wigman, Igor
(författare)
-
Fluctuations of the nodal length of random spherical harmonics
- 2010
-
Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 298:3, s. 787-831
-
Tidskriftsartikel (refereegranskat)abstract
- Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.
|
|
