5,472 research outputs found
The Defect of Random Hyperspherical Harmonics
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the
unit -sphere (). 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 2011) and a Central Limit Theorem (Marinucci and
Wigman 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-It\^o 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. Our argument requires some novel technical results of
independent interest that involve integrals of the product of three
hyperspherical harmonics.Comment: Accepted for publication in Journal of Theoretical Probabilit
Gaussian Approximations of Multiple Integrals
Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional
vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian
process. We establish necessary and sufficient conditions to have that, as l
diverges, the law of I(l) is asymptotically close (for example, in the sense of
Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the
same covariance matrix as I(l). The main feature of our results is that they
require minimal assumptions (basically, boundedness of variances) on the
asymptotic behaviour of the variances and covariances of the elements of I(l).
In particular, we will not assume that the covariance matrix of I(l) is
convergent. This generalizes the results proved in Nualart and Peccati (2005),
Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in
Marinucci and Peccati (2007b), the criteria established in this paper are
crucial in the study of the high-frequency behaviour of stationary fields
defined on homogeneous spaces.Comment: 15 page
Generating Functions for Coherent Intertwiners
We study generating functions for the scalar products of SU(2) coherent
intertwiners, which can be interpreted as coherent spin network evaluations on
a 2-vertex graph. We show that these generating functions are exactly summable
for different choices of combinatorial weights. Moreover, we identify one
choice of weight distinguished thanks to its geometric interpretation. As an
example of dynamics, we consider the simple case of SU(2) flatness and describe
the corresponding Hamiltonian constraint whose quantization on coherent
intertwiners leads to partial differential equations that we solve.
Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2)
flatness on coherent spin networks for arbitrary graphs.Comment: 31 page
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