1,720,985 research outputs found
The Defect of Random Hyperspherical Harmonics
No full textRandom hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-sphere (d≥ 2). We investigate the distribution of their defect, i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman in J Phys A Math Theor 44:355206, 2011) and a central limit theorem (Marinucci and Wigman in Commun Math Phys 327(3):849–872, 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener–Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein–Malliavin approximation techniques by Nourdin and Peccati (in Prob Theory Relat Fields 145(1–2):75–118, 2009; Normal approximations with Malliavin calculus. Cambridge Tracts in Mathematics, vol 192, Cambridge University Press, Cambridge, 2012). Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics
Asymptotic distribution of nodal intersections for arithmetic random waves
Full text linkWe study the nodal intersections number of random Gaussian toral Laplace eigenfunctions ('arithmetic random waves') against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for 'generic' curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar 'static' curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For 'generic' curves we prove the central limit theorem (at least, for 'most' of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues
Fluctuations of Level Curves for Time-Dependent Spherical Random Fields
Full text linkThe investigation of the behaviour for geometric functionals of random fields
on manifolds has drawn recently considerable attention. In this paper, we extend this framework
by considering fluctuations over time for the level curves of general isotropic Gaussian spherical
random fields. We focus on both long and short memory assumptions; in the former case, we
show that the fluctuations of u-level curves are dominated by a single component, corresponding
to a second-order chaos evaluated on a subset of the multipole components for the random field
Non-universal moderate deviation principle for the nodal length of arithmetic Random Waves
Full text linkInspired by the recent work Macci et al. (2021), we prove a non-universal non-central
Moderate Deviation Principle for the nodal length of arithmetic random waves (Gaussian Laplace
eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established by Marinucci et al. (2016) and
Benatar et al. (2020) respectively, by means of chaotic expansions, number theoretical estimates
and full correlation phenomena. Our proof is simple and relies on the interplay between the long
memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well
as on well-known techniques in Large Deviation theory (the contraction principle and the concept
of exponential equivalence)
Quantitative limit theorems for local functionals of arithmetic random waves
Full text linkWe consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz et al. (Ann Inst Fourier (Grenoble) 58(1):299–335, 2008), Krishnapur et al. (Ann Math 177(2):699–737, 2013), and Marinucci et al. (Geom Funct Anal 26(3):926–960, 2016). Our techniques involve Wiener-Itô chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula
Sojourn functionals of time-dependent -random fields on two-point homogeneous spaces
No full textWe investigate geometric properties of invariant spatio-temporal random fields X : Md ×
R → R defined on a compact two-point homogeneous space Md in any dimension
d ≥ 2, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as T → +∞) of the average on [0,T] of
the volume of the excursion set on the manifold, i.e. of {X(·, t) ≥ u} (for any u > 0).
The Fourier components of X may have short or long memory in time, i.e. integrable
or non-integrable temporal covariance functions. Our argument follows the approach
developed in Marinucci et al. (2021) and allows us to extend their results for invariant
spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chisquared distributed fields on two-point homogeneous spaces in any dimension. We find
that both the asymptotic variance and limiting distribution, as T → +∞, of the average
empirical volume turn out to be non-universal, depending on the memory parameters of
the field X
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