1,720,972 research outputs found
On the area of excursion sets of spherical Gaussian eigenfunctions
No full textThe 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.
© 2011 American Institute of Physic
The defect variance of random spherical harmonics
No full textThe defect of a function [IMAGE] is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is evaluating the defect variance, asymptotically in the high-frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern cosmological data analysis
On nonlinear functionals of random spherical eigenfunctions
No full textWe prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment
Fluctuations of the Euler-Poincare` Characteristic for Random Spherical Harmonics
Full text linkIn this short note, we build upon recent results from [7] to present a precise expression for the
asymptotic variance of the Euler-Poincar ́e characteristic for the excursion sets of Gaussian eigenfunctions
on S
2
; this result can be written as a second-order Gaussian kinematic formula for the excursion sets of
random spherical harmonics. The covariance between the Euler-Poincar ́e characteristics for different level
sets is shown to be fully degenerate; it is also proved that the variance for the zero level excursion sets is
asymptotically of smaller orde
Planck-scale distribution of nodal length of arithmetic random waves
Full text linkWe study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral quasi-correlations, i.e., unusually small sums of lattice points lying on the same circle
The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics
Full text linkWe study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics f(l) of high degree l -> infinity i.e. the length of their zero set f(l)(-1)(0). It is found that the nodal lengths are asymptotically equivalent, in the L-2-sense, to the "sample trispectrum", i.e., the integral of H-4(f(l)(x)), the fourth-order Hermite polynomial of the values of f(l). A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit
On the Distribution of the Critical Values of Random Spherical Harmonics
Full text linkWe 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
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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