Abstract
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-sphere (\(d\ge 2\)). 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.
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Notes
Recall that \(\left( P_\ell ^{(a,b)} \right) _\ell \) is a family of orthogonal polynomials on the interval \([-\,1,1]\) with respect to the weight \( (1-t)^{a}(1+t)^b \).
See [30, (3.1.3)]
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Acknowledgements
This topic was suggested by Domenico Marinucci. The author would like to thank him, Giovanni Peccati and Igor Wigman for useful conversations, and an anonymous referee for insightful comments. The research leading to this work was carried out within the framework of the ERC Pascal Project No. 277742 and of the Grant STARS (R-AGR-0502-10) at Luxembourg University. The author is currently supported by the Foundation Sciences Mathématiques de Paris and the ANR-17-CE40-0008 Project Unirandom.
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Appendix
Appendix
1.1 Proof of Lemma 2.1
Proof
By (1.7), one deduces that \(|D_\ell |\le |{\mathbb {S}}^d|\) a.s. and hence \(D_\ell \in L^2({\mathbb {P}})\). Recall that we can write
The chaotic expansion Sect. 3.2 of the indicator function \(1_{(0,+\infty )}\) is given by (see, e.g., [24] and the references therein)
where \(\phi \) still denotes the p.d.f. of the standard Gaussian law and \((H_k)_{k\ge 0}\) the sequence of Hermite polynomials. Hence in particular,
\(\Phi \) still denoting the cumulative distribution function of a standard Gaussian random variable. Actually, it is easy to check that, for \(Z\sim {\mathcal {N}}(0,1)\), \({\mathbb { E}}[1_{(0,+\infty )}(Z)] = 1/2\), whereas for \(k\ge 1\)
which vanishes if k is even. For \(m\in {\mathbb {N}}\), \(m>0\), let us consider the random variable
where the sum starts from \(q=1\) since \(H_1(t) = t\) and hyperspherical harmonics have zero mean over \({\mathbb {S}}^d\). Let us set moreover
In what follows, we shall show that the sequence of random variables \(\left( U_\ell ^m\right) _m\) is a Cauchy sequence in \(L^2({\mathbb {P}})\). By the orthogonality property of chaotic projections and (3.4), we have for \(m,n \in {\mathbb {N}}\), \(n,m>0\)
Now, since Gegenbauer polynomials are uniformly bounded by 1, we have
hence (7.1) allows to conclude the proof. \(\square \)
1.2 Some Useful Estimates
Let us denote \(H:=L^2({\mathbb {S}}^d)\).
Lemma 7.1
There exists \(C>0\) such that for integers \(q,p\ge 1\), \(q\le p\),
where
Proof
Recall that \(h_{\ell ;2q+1,d}\) can be expressed as a multiple Wiener–Itô integral of order q (see Sect. 3.2.1)
where the function \(g_{\ell ;2q+1,d}\) is given by
Similar arguments as those in the proof of [8, Lemma 6.1] allow one to have, for integers \(p,q\ge 1\), the following new estimates
and moreover for \(q\le p\)
where \(n:=2q+2p+2-2r\) for notational simplicity. Now from [21, Proposition 4.1] we know the explicit formula for the norm of contractions: for \(q\le p\)
where \(d\underline{x}:=dx_1dx_2dx_3dx_4\). Thanks to [21, Proposition 4.2, Proposition 4.3] (for \(q\ge 2\)) and Lemma 1.4 (for \(q=1\)) we have, as \(\ell \rightarrow +\infty \),
where \(R_{\ell ;2} = \log \ell / \ell ^{9/2}\) and \(R_{\ell ;d}=1/\ell ^{2d +(d-1)/2}\) for \(d>2\). Note that O’ notation is independent of q and p.
As stated in [8, (6.1),(6.2)], the following inequalities hold
Plugging (7.6) and (7.5) into (7.3) and (7.4), one infers (7.2). \(\square \)
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Rossi, M. The Defect of Random Hyperspherical Harmonics. J Theor Probab 32, 2135–2165 (2019). https://doi.org/10.1007/s10959-018-0849-6
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DOI: https://doi.org/10.1007/s10959-018-0849-6
Keywords
- Defect
- Gaussian eigenfunctions
- High-energy asymptotics
- Quantitative central limit theorem
- Integrals of hyperspherical harmonics