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.

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

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

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. $\square$

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 $q,p\ge 1$, $q\le p$,

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

(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). $\square$