Quantitative Limit Theorems for Local Functionals of Arithmetic Random Waves

Abstract

We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz et al. (Ann Inst Fourier (Grenoble) 58(1):299–335, 2008), Krishnapur et al. (Ann Math 177(2):699–737, 2013), and Marinucci et al. (Geom Funct Anal 26(3):926–960, 2016). Our techniques involve Wiener-Itô chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula.

Appendix

Proof (Lemma 3 )

From [13, Lemma 3.4], we have that the chaotic expansion of $\mathscr Z_n^{\,\,\varepsilon }$ is

$$\displaystyle \begin{aligned} \mathscr Z_n^{\,\,\varepsilon} = \sum_{q=0}^{+\infty} \mathscr Z_n^{\,\,\varepsilon}[2q] = \sum_{q=0}^{+\infty} \frac{\beta^\varepsilon_{2q}}{(2q)!} \int_{\mathbb T} H_{2q}(T_n(x))\,dx, \end{aligned} $$

(79)

where H _2q denotes the 2q-th Hermite polynomial, and

$$\displaystyle \begin{aligned} \beta^\varepsilon_{0} = \frac{1}{2\varepsilon}\int_{-\varepsilon}^\varepsilon \phi(t)\,dt,\qquad \beta^\varepsilon_{2q} = -\frac{1}{\varepsilon} \phi(\varepsilon) H_{2q-1}(\varepsilon),\ q\ge 1,\end{aligned} $$

(80)

ϕ still denoting the Gaussian density. Taking the limit for ε going to 0 in (80) we obtain the collection of coefficients (37), related to the (formal) Hermite expansion of the Dirac mass δ ₀. Note that

$$\displaystyle \begin{aligned} \sum_{q=1}^{+\infty} \frac{(\beta_{2q})^2}{(2q)!} \int_{\mathbb T} r_n(x)^{2q}\,dx = \frac{1}{2\pi} \int_{\mathbb T} \left( \frac{1}{\sqrt{1-r_n(x)^2}} -1 \right)\,dx < +\infty,\end{aligned} $$

(81)

since the collection {(β _2q)²∕(2q)!}_q coincides with the sequence of Taylor coefficients of the function $x\mapsto 1/(2\pi \sqrt {1 - x^2})$ around zero; thanks to Lemma 5.3 in [15] we have the finiteness of the integral. Therefore the series

$$\displaystyle \begin{aligned} \sum_{q=0}^{+\infty} \frac{\beta_{2q}}{(2q)!} \int_{\mathbb T} H_{2q}(T_n(x))\,dx,\end{aligned} $$

is a well-defined random variable in $L^2(\mathbb P)$, its variance being the series on the l.h.s. of (81). Moreover, from [1, 22.14.16] and (81)

$$\displaystyle \begin{aligned} \sum_{q=1}^{+\infty} \frac{(\beta_{2q}^\varepsilon - \beta_{2q})^2}{(2q)!} \int_{\mathbb T} r_n(x)^{2q}\,dx\le 2 \sum_{q=1}^{+\infty} \frac{(\beta_{2q})^2}{(2q)!} \int_{\mathbb T} r_n(x)^{2q}\,dx <+\infty,\end{aligned} $$

that implies, by the dominated convergence theorem, $\mathscr Z_n^\varepsilon \mathop {\to } \mathscr Z_n,$ ε → 0, in $L^2(\mathbb P)$. □

Proof (Lemma 4 )

From (44) with q = 2

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathscr L_n[4] = \frac{\sqrt{E_n}}{128 \sqrt{2}}\Big (8 \int_{\mathbf{T}} H_4(T_n(x))\,dx - \int_{\mathbf{T}} H_4(\widetilde \partial_1 T_n(x))\,dx - \int_{\mathbf{T}} H_4(\widetilde \partial_2 T_n(x))\,dx \\ - 8 \int_{\mathbf{T}} H_2(T_n(x)) H_2(\widetilde \partial_1 T_n(x))\,dx - 8 \int_{\mathbf{T}} H_2(T_n(x)) H_2(\widetilde \partial_2 T_n(x))\,dx \\ -2 \int_{\mathbf{T}} H_2(\widetilde \partial_1 T_n(x))H_2(\widetilde \partial_2 T_n(x))\,dx.\vspace{2pt} {} \end{array} \end{aligned} $$

Lemmas 5.2 and 5.5 in [13] together with some straightforward computations allow one to write, from (82),

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathscr L_n[4] = \sqrt{\frac{E_n}{\mathscr N_n^2}}\frac{1}{128 \sqrt{2}}\Big ( 8 W_1^2 - 16 W_2^2 -16 W_3^2 -32 W_4^2 \\ + \frac{1}{\mathscr N_n} \sum_{\lambda\in \varLambda_n} |a_\lambda|{}^4 \left(-8 +12 \left(\left( \frac{\lambda_1}{\sqrt{n}}\right)^2 + \left( \frac{\lambda_2}{\sqrt{n}}\right)^2 \right)^2 \right)\Big ).\vspace{-3pt} {} \end{array} \end{aligned} $$

Recalling that $\lambda _1^2 + \lambda _2 ^2 = n$, we obtain (47). Let us now note that we can write

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} W_1^2 - 2W_2^2 -2W_3^2 -4W_4^2 \\ = \frac{1}{\mathscr N_n/2}\sum_{\lambda, \lambda'\in \varLambda_n^+} \left ( 1 -\frac{2}{n^2}(\lambda_1 \lambda_1^{\prime} + \lambda_2 \lambda_2^{\prime})^2 \right )(|a_\lambda |{}^2 -1) (|a_{\lambda^{\prime}} |{}^2 -1).\vspace{-3pt} \end{array} \end{aligned} $$

(82)

Then it is immediate to compute from (82)

$$\displaystyle \begin{aligned} \mathbb E\left[W_1^2 - 2W_2^2 -2W_3^2 -4W_4^2 \right] = -1. \end{aligned} $$

(83)

Bearing in mind Lemma 4.1 in [13], still from (82) some straightforward computations lead to

$$\displaystyle \begin{aligned} \mathbb E\left[(W_1^2 - 2W_2^2 -2W_3^2 -4W_4^2 )^2 \right] = 2 + \widehat{\mu_n}(4)^2 +\frac{48}{\mathscr N_n}. \end{aligned} $$

(84)

From (83) and (84) hence we find

$$\displaystyle \begin{aligned} \text{Var}(W_1^2 - 2W_2^2 -2W_3^2 -4W_4^2)= 1 + \widehat{\mu_n}(4)^2 +\frac{48}{\mathscr N_n}.\end{aligned} $$

Recalling that $(\sqrt {2} |a_\lambda |)^2$ is distributed as a chi-square random variable with two degrees of freedom,

$$\displaystyle \begin{aligned} \text{Var}\left( \frac{1}{2} \frac{1}{\mathscr N_n}\sum_{\lambda\in \varLambda_n} |a_\lambda|{}^4 \right) =\frac{10}{\mathscr N_n},\end{aligned} $$

(85)

and moreover

$$\displaystyle \begin{aligned} \text{Cov}\left(W_1^2 - 2W_2^2 -2W_3^2 -4W_4^2, \frac{1}{2} \frac{1}{\mathscr N_n}\sum_{\lambda\in \varLambda_n} |a_\lambda|{}^4 \right) =-\frac{12}{\mathscr N_n}. \end{aligned}$$

This concludes the proof of Lemma 4. □