The Defect of Random Hyperspherical Harmonics

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

  1. 1.

    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 \( (1-t)^{a}(1+t)^b \).

  2. 2.

    See [30, (3.1.3)]

References

  1. 1.

    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York (2007)

    Google Scholar 

  2. 2.

    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  3. 3.

    Benatar, J., Marinucci, D., Wigman, I.: Planck-scale distribution of nodal length of arithmetic random waves. arXiv:1710.06153

  4. 4.

    Berry, M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A: Math. Theor. 10(12), 2083–2091 (1977)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Berry, M.V.: Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A: Math. Gen. 35(13), 3025–3038 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Brüning, J., Gromes, D.: Über die Länge der Knotenlinien schwingender Membranen. Math. Z. 124, 79–82 (1972)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Brüning, J.: Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators. Math. Z. 158(1), 15–21 (1978)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cammarota, V., Marinucci, D.: On the limiting behaviour of needlets polyspectra. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 51(3), 1159–1189 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cammarota, V., Marinucci, D.: A quantitative central limit theorem for the Euler-Poincaré characteristic of random spherical eigenfunctions. Ann. Prob. (in press)

  10. 10.

    Cammarota, V., Marinucci, D., Wigman, I.: Fluctuations of the Euler–Poincaré characteristic for random spherical harmonics. Proc. Am. Math. Soc. 144(11), 4759–4775 (2016)

    Article  Google Scholar 

  11. 11.

    Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Durastanti, C.: Adaptive global thresholding on the sphere. J. Multivar. Anal. 151, 110–132 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Faraut, J.: Analysis on Lie groups. Cambridge Studies in Advanced Mathematics, vol. 110. Cambridge University Press, Cambridge (2008). An introduction

    Google Scholar 

  14. 14.

    Ghosh, A., Reznikov, A., Sarnak, P.: Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5), 1515–1568 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Jung, J., Zelditch, S.: Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary. Math. Ann. 364(3–4), 813–840 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Krishnapur, M., Kurlberg, P., Wigman, I.: Nodal length fluctuations for arithmetic random waves. Ann. Math. (2) 177(2), 699–737 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Marinucci, D.: High-resolution asymptotics for the angular bispectrum of spherical random fields. Ann. Stat. 34(1), 1–41 (2006)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Marinucci, D.: A central limit theorem and higher order results for the angular bispectrum. Probab. Theory Relat. Fields 141(3–4), 389–409 (2008)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Marinucci, D., Peccati, G.: Random Fields on the Sphere. London Mathematical Society Lecture Note Series, vol. 389. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  20. 20.

    Marinucci, D., Peccati, G., Rossi, M., Wigman, I.: Non-Universality of nodal lengths distribution for arithmetic random waves. Geom. Funct. Anal. 26(3), 926–960 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Marinucci, D., Rossi, M.: Stein–Malliavin approximations for nonlinear functionals of random eigenfunctions on \(\mathbb{S}^d\). J. Funct. Anal. 268(8), 2379–2420 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Marinucci, D., Rossi, M., Wigman, I.: The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics. arXiv:1705.05747

  23. 23.

    Marinucci, D., Wigman, I.: The defect variance of random spherical harmonics. J. Phys. A: Math. Theor. 44, 355206 (2011)

    Article  Google Scholar 

  24. 24.

    Marinucci, D., Wigman, I.: On nonlinear functionals of random spherical eigenfunctions. Commun. Math. Phys. 327(3), 849–872 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Meckes, E.: On the approximate normality of eigenfunctions of the Laplacian. Trans. Am. Math. Soc. 361(10), 5377–5399 (2009)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Nazarov, F., Sodin, M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145(1–2), 75–118 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus. Cambridge Tracts in Mathematics, vol. 192. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  29. 29.

    Peccati, G., Tudor, C.A.: Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII, volume 1857 of Lecture Notes in Mathematics, pp. 247–262. Springer, Berlin (2005)

    Google Scholar 

  30. 30.

    Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series, vol. 1. Springer, Bocconi University Press, Milan (2011)

    Google Scholar 

  31. 31.

    Pham, V.-H.: On the rate of convergence for central limit theorems of sojourn times of Gaussian fields. Stoch. Process. Appl. 123(6), 2158–2174 (2013)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rossi, M.: The geometry of spherical random fields. Ph.D. thesis. University of Rome Tor Vergata (2015). arXiv:1603.07575

  33. 33.

    Rudnick, Z., Wigman, I.: On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9(1), 109–130 (2008)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Szegő, G.: Orthogonal polynomials, vol. XXIII, 4th edn. American Mathematical Society, Providence, RI (1975)

    Google Scholar 

  35. 35.

    Vilenkin, N.J., Klimyk, A.U.: Representation of Lie groups and special functions. Volume 74 of Mathematics and its Applications (Soviet Series), vol. 2. Kluwer Academic Publishers Group, Dordrecht (1993)

    Google Scholar 

  36. 36.

    Varshalovich, D.A., Moskalev, A.N., Khersonskiĭ, V.K.: Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Teaneck (1988)

    Google Scholar 

  37. 37.

    Wigman, I.: Fluctuations of the nodal length of random spherical harmonics. Commun. Math. Phys. 298(3), 787–831 (2010)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wigman, I.: On the Nodal Lines of Random and Deterministic Laplace Eigenfunctions. Spectral geometry, Volume 84 of Proceedings of the International Conference on Spectral Geometry, Dartmouth College, pp. 285–297. American Mathematical Society, Providence (2012)

    Google Scholar 

  39. 39.

    Yau, S.-T.: Survey on partial differential equations in differential geometry. Seminar on Differential Geometry, Volume 102 of Annals of Mathematical Studies, pp. 3–71. Princeton University Press, Princeton (1982)

    Google Scholar 

Download references

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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Correspondence to Maurizia Rossi.

Appendix

Appendix

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

$$\begin{aligned} D_\ell = 2\int _{{\mathbb {S}}^d} 1_{(0,+\infty )}(T_\ell (x))\,\hbox {d}x - |{\mathbb {S}}^d|. \end{aligned}$$

The chaotic expansion Sect. 3.2 of the indicator function \(1_{(0,+\infty )}\) is given by (see, e.g., [24] and the references therein)

$$\begin{aligned} 1_{(0,+\infty )}(\cdot ) = \frac{1}{2} + \sum _{q\ge 0} \frac{\phi (0)H_{2q}(0)}{(2q+1)!} H_{2q+1}(\cdot ), \end{aligned}$$

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,

$$\begin{aligned} \sum _{q\ge 0} \frac{(\phi (0)H_{2q}(0))^2}{(2q+1)!} = \Phi (0)(1-\Phi (0))< +\infty , \end{aligned}$$
(7.1)

\(\Phi \) 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 \(k\ge 1\)

$$\begin{aligned} \begin{aligned} {\mathbb { E}}[1_{(0,+\infty )}(Z)H_{k}(Z)]&=\int _0^{+\infty } (-1)^k \phi ^{-1}(t)\frac{\hbox {d}^k\phi }{\hbox {d}t^k}(t)\phi (t)\,\hbox {d}t\\&= (-1)^k \frac{d^{k-1}\phi }{\mathrm{d}t^{k-1}}(t) \Big |_0^{+\infty } = -\phi (t) H_{k-1}(t) \Big |_0^{+\infty } \\&=\phi (0) H_{k-1}(0), \end{aligned} \end{aligned}$$

which vanishes if k is even. For \(m\in {\mathbb {N}}\), \(m>0\), let us consider the random variable

$$\begin{aligned} \begin{aligned} U_\ell ^m&:= 2\int _{{\mathbb {S}}^d} \left( \frac{1}{2} + \sum _{q= 0}^m \frac{\phi (0)H_{2q}(0)}{(2q+1)!} H_{2q+1}(T_\ell (x) ) \right) \hbox {d}x - |{\mathbb {S}}^d|\\&=\sum _{q= 1}^m 2\frac{\phi (0)H_{2q}(0)}{(2q+1)!}\int _{{\mathbb {S}}^d} H_{2q+1}(T_\ell (x) )\,\hbox {d}x, \end{aligned} \end{aligned}$$

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

$$\begin{aligned} J_{2q+1} := 2\phi (0)H_{2q}(0). \end{aligned}$$

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 \(m,n \in {\mathbb {N}}\), \(n,m>0\)

$$\begin{aligned} \begin{aligned} {\mathbb { E}}[(U_\ell ^m - U_\ell ^{m+n})^2] = \sum _{q=m+1}^{m+n} \frac{J_{2q+1}^2}{(2q+1)!}\int _{({\mathbb {S}}^d)^2} G_{\ell ;d}(\cos d(x,y))^{2q+1}\,\hbox {d}x\hbox {d}y. \end{aligned} \end{aligned}$$

Now, since Gegenbauer polynomials are uniformly bounded by 1, we have

$$\begin{aligned} \begin{aligned} {\mathbb { E}}[(U_\ell ^m - U_\ell ^{m+n})^2]&\le |\mathbb S^d|^2\sum _{q=m+1}^{m+n} \frac{J_{2q+1}^2}{(2q+1)!}; \end{aligned} \end{aligned}$$

hence (7.1) allows to conclude the proof. \(\square \)

Some Useful Estimates

Let us denote \(H:=L^2({\mathbb {S}}^d)\).

Lemma 7.1

There exists \(C>0\) such that for integers \(q,p\ge 1\), \(q\le p\),

$$\begin{aligned} \begin{aligned}&\mathrm{Var}\left( \left\langle D h_{\ell ;2q+1,d} , -DL^{-1} h_{\ell ;2q+1,d} \right\rangle _H \right) \le C (2q+1)^2 ((2q)!)^2 3^{4q} R_{\ell ;d},\\&{\mathbb { E}}\left[ \left\langle D h_{\ell ;2q+1,d} , -DL^{-1} h_{\ell ;2p+1,d} \right\rangle _H ^2 \right] \le C (2q+1)^2(2q)! (2p)! 3^{2q+2p} R_{\ell ;d}, \end{aligned} \end{aligned}$$
(7.2)

where

$$\begin{aligned} R_{\ell ;2} := \frac{\log \ell }{\ell ^{9/2}}\quad \text { and for } d>2 \quad R_{\ell ;d}:=\frac{1}{\ell ^{2d +(d-1)/2}}. \end{aligned}$$

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)

$$\begin{aligned} h_{\ell ;2q+1,d} \mathop {=}^{{\mathcal {L}}} \int _{({\mathbb {S}}^d)^q} g_{\ell ;2q+1,d}(y_1,y_2,\dots ,y_q)\,\hbox {d}W(y_1)\hbox {d}W(y_2)\dots \hbox {d}W(y_q)=:I_q(g_{\ell ;2q+1,d}), \end{aligned}$$

where the function \(g_{\ell ;2q+1,d}\) is given by

$$\begin{aligned} g_{\ell ;2q+1,d}(y_1,y_2,\dots ,y_q) := \int _{{\mathbb {S}}^d} \left( \frac{n_{\ell ;d}}{|{\mathbb {S}}^d|} \right) ^{q/2} G_{\ell ;d}(\cos d(x,y_1)) \cdots G_{\ell ;d}(\cos d(x,y_q))\,\hbox {d}x. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\mathrm{Var}\left( \left\langle D h_{\ell ;2q+1,d} , -DL^{-1} h_{\ell ;2q+1,d} \right\rangle _H \right) \\&\quad \le (2q+1)^2 \sum _{r=1}^{2q} ((r-1)!)^2 { 2q \atopwithdelims ()r-1}^4 (2(2q+1)-2r)!\Vert g_{\ell ;2q+1,d}\\&\qquad \otimes _r g_{\ell ;2q+1,d}\Vert ^2_{H^{\otimes 2(2q+1)-2r}}, \end{aligned} \end{aligned}$$
(7.3)

and moreover for \(q\le p\)

$$\begin{aligned} \begin{aligned}&{\mathbb { E}}\left[ \left\langle D h_{\ell ;2q+1,d} , -DL^{-1} h_{\ell ;2p+1,d} \right\rangle _H ^2 \right] \\&\quad = (2q+1)^2 \sum _{r=1}^{2q+1} ((r-1)!)^2 { 2q \atopwithdelims ()r-1}^2 { 2p \atopwithdelims ()r-1}^2\\&\qquad \times (2q+2p + 2-2r)!\Vert g_{\ell ;2q+1,d} {\widetilde{\otimes }}_r g_{\ell ;2p+1,d}\Vert ^2_{H^{\otimes n}} \\&\quad \le (2q+1)^2 \sum _{r=1}^{2q+1} ((r-1)!)^2 { 2q \atopwithdelims ()r-1}^2 { 2p \atopwithdelims ()r-1}^2\\&\qquad \times (2q+2p + 2-2r)!\Vert g_{\ell ;2q+1,d} \otimes _r g_{\ell ;2p+1,d}\Vert ^2_{H^{\otimes n}}, \end{aligned} \end{aligned}$$
(7.4)

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 \(q\le p\)

$$\begin{aligned}&\Vert g_{\ell ;2q+1,d} \otimes _r g_{\ell ;2p+1,d}\Vert ^2_{H^{\otimes n}} =\int _{({\mathbb {S}}^d)^4}G_{\ell ;d}(\cos d(x_1,x_2))^r G_{\ell ;d}(\cos d(x_2,x_3))^{2q+1-r} \\&\quad \times G_{\ell ;d}(\cos d(x_3,x_4))^r G_{\ell ;d}(\cos d(x_4,x_1))^{2q+1-r}\hbox {d}\underline{x}, \end{aligned}$$

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 \(\ell \rightarrow +\infty \),

$$\begin{aligned} \Vert g_{\ell ;2q+1,2} \otimes _r g_{\ell ;2p+1,2}\Vert ^2_{H^{\otimes n}} = O\left( R_{\ell ;d} \right) , \end{aligned}$$
(7.5)

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

$$\begin{aligned}&\sum _{r=1}^{2q} ((r-1)!)^2 { 2q \atopwithdelims ()r-1}^4 (2(2q+1)-2r)!\le ((2q)!)^2 3^{4q},\nonumber \\&\sum _{r=1}^{2q+1} ((r-1)!)^2 { 2q \atopwithdelims ()r-1}^2 { 2p \atopwithdelims ()r-1}^2 (2q+2p + 2-2r)!\le (2q)! (2p)! 3^{2q+2p}.\nonumber \\ \end{aligned}$$
(7.6)

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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Keywords

  • Defect
  • Gaussian eigenfunctions
  • High-energy asymptotics
  • Quantitative central limit theorem
  • Integrals of hyperspherical harmonics

Mathematics Subject Classification (2010)

  • 60G60
  • 42C10
  • 60D05
  • 60B10
  • 43A75