Functional Convergence of Berry’s Nodal Lengths: Approximate Tightness and Total Disorder

Abstract

We consider Berry’s random planar wave model (J Phys A 10(12):2083–2092, 1977), and prove spatial functional limit theorems—in the high-energy limit—for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zitikis (Ann Inst Stat Math 60(2):345–365, 2008).

Appendices

Appendix A: Proof of Proposition 1.3

According to Proposition 1.2, the random field $X_E$ converges in the sense of finite-dimensional distributions to $\textbf{W}$, and moreover one has that

$$\begin{aligned} \sup _{E>0, \, \textbf{t}\in [0,1]^2} {\mathbb {E}}\left[ X_E(\textbf{t})^2\right] <\infty , \quad \text{ and } \quad {\mathbb {E}}\left[ X_E(\textbf{t})^2\right] \rightarrow {\mathbb {E}}\left[ \textbf{W}(\textbf{t})^2\right] , \end{aligned}$$

where the first relation follows from the computations contained in [31, Sections 6 and 7], and the second one takes place as $E\rightarrow \infty$, for all $\textbf{t}\in [0,1]^2$. As a consequence of these relations, we can apply [20, Theorem 4] and conclude that, if $\psi \in {\mathcal {C}}_c^\infty (R)$, then

$$\begin{aligned} \int _R X_E(\textbf{t}) \psi (\textbf{t}) d\textbf{t} \xrightarrow []{d} \int _R \textbf{W}(\textbf{t}) \psi (\textbf{t}) d\textbf{t}. \end{aligned}$$

(A.1)

We now fix $\varphi \in {\mathcal {C}}_c^\infty (R)$ and apply (A.1) to $\psi (\textbf{t}) = \frac{\partial }{\partial t_1}\frac{\partial }{\partial t_2}\varphi (\textbf{t})\in {\mathcal {C}}_c^\infty (R)$, where $\textbf{t} = (t_1,t_2)$, in such a way that $\varphi (\textbf{t}) = \int _{(t_1, 1)\times (t_2, 1)} \psi (\textbf{z}) d\textbf{z}$. Applying a standard Fubini theorem on the left-hand side of (A.1) and a stochastic Fubini theorem (see [36, Theorem 5.13.1]) on the right-hand side yields that

$$\begin{aligned} \int _R X_E(\textbf{t}) \psi (\textbf{t}) d\textbf{t} = \langle \widetilde{{\mathscr {L}}}_E, \varphi \rangle , \quad \text{ and } \quad \int _R \textbf{W}(\textbf{t}) \psi (\textbf{t}) d\textbf{t} = \int _R \varphi (\textbf{z}) \textbf{W}(d\textbf{z}), \end{aligned}$$

where the last expression denotes a stochastic Wiener-Itô integral with respect to $\textbf{W}$. The conclusion now follows from [16, Theorem III.6.5].

Appendix B: Proof of Lemma 1.4

Since $U_n$ and $V_n$ converge weakly to X and zero in ${\textbf{D}}_2$, respectively, we use for instance [44, Theorem 2], to deduce that, for every $\varepsilon >0$,

$$\begin{aligned} \lim _{\delta \rightarrow 0} \limsup _{n\rightarrow \infty } {\mathbb {P}}\left\{ \omega _{\delta }(U_n)>\varepsilon \right\} = 0, \quad \lim _{\delta \rightarrow 0} \limsup _{n\rightarrow \infty } {\mathbb {P}}\left\{ \omega _{\delta }(V_n)>\varepsilon \right\} = 0 \end{aligned}$$

(B.1)

where $\omega _{\delta }(f):= \sup \left\{ | f(\textbf{t}) - f(\textbf{s})|: \Vert \textbf{t}-\textbf{s}\Vert <\delta \right\}$. To obtain the desired conclusion, by virtue of the discussion contained in [29, p.1291], it is sufficient to show that

$$\begin{aligned} \lim _{\delta \rightarrow 0} \limsup _{n\rightarrow \infty } {\mathbb {P}}\left\{ \omega _{\delta }(X_n)>\varepsilon \right\} = 0. \end{aligned}$$

By the triangle inequality, we can write for every $\delta >0$,

$$\begin{aligned} \omega _{\delta }(X_n) = \omega _{\delta }(U_n+V_n+W_n) \le \omega _{\delta }(U_n) +\omega _{\delta }(V_n)+ \omega _{\delta }(W_n), \end{aligned}$$

in such a way that

$$\begin{aligned} {\mathbb {P}}\left\{ \omega _{\delta }(X_n)>\varepsilon \right\} \le {\mathbb {P}}\left\{ \omega _{\delta }(U_n)>\varepsilon /3\right\} + {\mathbb {P}}\left\{ \omega _{\delta }(V_n)>\varepsilon /3\right\} + {\mathbb {P}}\left\{ \omega _{\delta }(W_n)>\varepsilon /3\right\} . \end{aligned}$$

Using the estimate $\omega _{\delta }(W_n) \le 2\sup _{\textbf{t}\in [0,1]^2} |W_n(\textbf{t})|$ and letting $n\rightarrow \infty$ and $\delta \rightarrow 0$ then implies the desired conclusion from (B.1) and assumption (iii) in the statement.

Appendix C: Moment Estimates for Suprema of Gaussian Fields

In what follows we consider a centred smooth stationary Gaussian field $G=\left\{ G(x):x\in {\mathbb {R}}^d\right\}$ on ${\mathbb {R}}^d$ with covariance function ${\mathbb {E}}\left[ G(x)G(y)\right] =\kappa (x-y)$. For an integer $j\ge 0$ and ${\mathcal {D}}\subset {\mathbb {R}}^d$, we write

$$\begin{aligned} \sigma ^2({\mathcal {D}};j):= \sup _{x\in {\mathcal {D}}} \sup _{|\alpha |\le j} {\mathbb {E}}\left[ (\partial _{\alpha } G(x))^2\right] , \end{aligned}$$

where $\partial _{\alpha }G(x):= \partial ^{\alpha _1}_{x_1}\ldots \partial ^{\alpha _d}_{x_d}G(x)$, for $\alpha :=(\alpha _1,\ldots , \alpha _d)$ with $|\alpha |:= \sum _{k=1}^d \alpha _k$. Moreover, for ${\mathcal {D}}\subset {\mathbb {R}}^d$ and $\varepsilon >0$, we write ${\mathcal {D}}^{(\varepsilon )}$ for the $\varepsilon$-enlargement of ${\mathcal {D}}$. Finally, we use the notation

$$\begin{aligned} \Vert f\Vert _{C^j({\mathcal {D}})}:= \sup _{x\in {\mathcal {D}}} \sup _{|\alpha |\le j} |\partial _{\alpha } f(x)| \end{aligned}$$

for $f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$. The goal of this section is to prove Proposition 3.6, whose statement we recall for convenience.

Proposition C.1

Let the above setting prevail. Assume that for every $m\ge 0$, there exists ${\tilde{\sigma }}^2(m)<\infty$ such that

$$\begin{aligned} {\mathbb {E}}\left[ (\partial _{\alpha } G(x))^2\right] \le {\tilde{\sigma }}^2(m), \quad \forall \alpha \in {\mathbb {N}}^d, |\alpha |\le m. \end{aligned}$$

(C.1)

Then, for every $p\ge 1$ and $j\ge 0$

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}^p\right] \le C \left\{ \log (\textrm{vol}({\mathcal {D}}))\right\} ^{p/2} \end{aligned}$$

where $C>0$ is an absolute constant depending on p and j, and $\textrm{vol}({\mathcal {D}})$ is the d-dimensional volume of ${\mathcal {D}}$.

We remark that assumption (C.1) in particular implies that $\sigma ^2({\mathcal {D}};j) \le {\tilde{\sigma }}^2(j)$ for every $j\ge 0$.

1.1 C.1: Proof of Proposition C.1

The proof of Proposition C.1 is based on several classical concentration inequalities for suprema of Gaussian fields, that we state here below. The first statement is an estimate for the first moment of $\Vert G\Vert _{C^j({\mathcal {D}})}$ (see [32, Appendix A.9]).

Proposition C.2

Let the above setting prevail.

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}\right] \le c_1({\mathcal {D}}) \sigma ({\mathcal {D}}^{(1)};j+1), \end{aligned}$$

(C.2)

where $c_1({\mathcal {D}})$ is a constant depending on ${\mathcal {D}}$.

The following inequality is the so-called Borell-TIS inequality applied to the Gaussian field $\partial _{\alpha }G$, see for instance [42, Theorem 2.1.1].

Proposition C.3

For every $\alpha \in {\mathbb {N}}^d$ and $u>0$, we have

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{x\in D} \partial _{\alpha } G(x) > {\mathbb {E}}\left[ \sup _{x\in {\mathcal {D}}} \partial _{\alpha } G(x)\right] + u\right\} \le e^{-\frac{u^2}{2 \sigma ^2({\mathcal {D}};|\alpha |)}}. \end{aligned}$$

(C.3)

Combining the contents of Propositions C.2 and C.3, we deduce that for every $\alpha \in {\mathbb {N}}^d$ with $|\alpha |\le j$ and $u>0$

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{x\in {\mathcal {D}}} \partial _{\alpha } G(x) > c_1({\mathcal {D}}) {\tilde{\sigma }}(j+1)+u \right\} \le e^{-\frac{u^2}{2{\tilde{\sigma }}^2(j+1)}} \end{aligned}$$

which implies (by symmetry)

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{x\in {\mathcal {D}}} |\partial _{\alpha } G(x)| > c_1({\mathcal {D}}) {\tilde{\sigma }}(j+1)+u \right\} \le 2e^{-\frac{u^2}{2{\tilde{\sigma }}^2(j+1)}}. \end{aligned}$$

Therefore summing over all possible $\alpha$ with $|\alpha |\le j$,

$$\begin{aligned} {\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathcal {D}})} > c_1({\mathcal {D}}) {\tilde{\sigma }}(j+1)+u \right\} \le k(j,d) e^{-\frac{u^2}{2\sigma ^2(j+1)}} \end{aligned}$$

(C.4)

where $k(j,d):=2\textrm{card}\left\{ \alpha \in {\mathbb {N}}^d: |\alpha |=j\right\}$.

We can now prove Proposition C.1.

Proof of Proposition C.1

By stationarity of G it follows that, if ${\mathcal {D}}'$ is a translation of ${\mathcal {D}}$, then necessarily $c_1({\mathcal {D}})=c_1({\mathcal {D}}')$, where $c_1({\mathcal {D}})$ is the constant appearing in (C.2). In particular, applying (C.2) in the case where ${\mathcal {D}}$ is a ball ${\mathbb {B}}$ with unit radius and exploiting the moment assumption (C.1) on G, we deduce that

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathbb {B}})}\right] \le c_1 {\tilde{\sigma }}(j+1), \end{aligned}$$

where $c_1$ is a universal constant. Therefore, applying (C.4) with ${\mathcal {D}}={\mathbb {B}}$ yields

$$\begin{aligned} {\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathbb {B}})}> c_1 {\tilde{\sigma }}(j+1)+u \right\} \le k(j,d) e^{-\frac{u^2}{2{\tilde{\sigma }}^2(j+1)}}, \quad u>0. \end{aligned}$$

Now, using the above inequality with $u=t-c_1 {\tilde{\sigma }}(j+1)$, we can write for every $b>0$ (setting $k:=k(j,d), {\tilde{\sigma }}:={\tilde{\sigma }}(j+1)$),

$$\begin{aligned} {\mathbb {E}}\left[ e^{b \Vert G\Vert _{C^j({\mathbb {B}})}}\right]= & {} 1+ b\int _{0}^{\infty } e^{tb}{\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathbb {B}})}> c_1 {\tilde{\sigma }} +(t-c_1 {\tilde{\sigma }}) \right\} dt \nonumber \\= & {} e^{b c_1 {\tilde{\sigma }}} + b\int _{c_1{\tilde{\sigma }}}^{\infty } e^{tb}{\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathbb {B}})} > c_1 {\tilde{\sigma }} +(t-c_1 {\tilde{\sigma }}) \right\} dt \nonumber \\\le & {} e^{b c_1 {\tilde{\sigma }}} + bk\int _{c_1{\tilde{\sigma }}}^{\infty } e^{tb} e^{-\frac{(t-c_1{\tilde{\sigma }})^2}{2{\tilde{\sigma }}^2}} dt \le e^{b c_1 {\tilde{\sigma }}} + bk\int _{{\mathbb {R}}} e^{tb} e^{-\frac{(t-c_1{\tilde{\sigma }})^2}{2{\tilde{\sigma }}^2}} dt \nonumber \\= & {} e^{b c_1 {\tilde{\sigma }}} + bk \sqrt{2\pi } {\tilde{\sigma }}{\mathbb {E}}\left[ e^{bZ}\right] , \quad Z \sim {\mathcal {N}}(c_1{\tilde{\sigma }},{\tilde{\sigma }}^2) \nonumber \\= & {} e^{b c_1 {\tilde{\sigma }}}+ bk \sqrt{2\pi } {\tilde{\sigma }}\left( e^{bc_1{\tilde{\sigma }}+b^2{\tilde{\sigma }}^2/2} \right) = e^{bc_1 {\tilde{\sigma }}} (1+bk\sqrt{2\pi } {\tilde{\sigma }}e^{b^2{\tilde{\sigma }}^2/2}) \nonumber \\\le & {} e^{bc_1 {\tilde{\sigma }} + b^2 {\tilde{\sigma }}^2/2}(1+bk\sqrt{2\pi } {\tilde{\sigma }}) \le e^{bc_1 {\tilde{\sigma }} + b^2 {\tilde{\sigma }}^2/2+bk\sqrt{2\pi } {\tilde{\sigma }}}\nonumber \\= & {} e^{b{\tilde{\sigma }}(c_1 +k\sqrt{2\pi } ) + b^2 {\tilde{\sigma }}^2/2}, \end{aligned}$$

(C.5)

where we used that $1+x\le e^x$. Now for ${\mathcal {D}}\subset {\mathbb {R}}^d$ we denote by $N_{{\mathcal {D}}}$ the minimal number of unit balls needed to cover ${\mathcal {D}}$ and by ${\mathcal {B}}_{{\mathcal {D}}}:=\left\{ {\mathbb {B}}_1,\ldots , {\mathbb {B}}_{N_{{\mathcal {D}}}}\right\}$ the collection of all unit balls covering ${\mathcal {D}}$ in such a way that $\textrm{card}({\mathcal {B}}_{{\mathcal {D}}})=N_{{\mathcal {D}}}$. Then, we have that, for every $b>0$

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}\right]= & {} {\mathbb {E}}\left[ \log \exp (b^{-1}b\Vert G\Vert _{C^j({\mathcal {D}})} )\right] = b^{-1} {\mathbb {E}}\left[ \log e^{b \Vert G\Vert _{C^j({\mathcal {D}})} }\right] \\\le & {} b^{-1} \log {\mathbb {E}}\left[ e^{b\Vert G\Vert _{C^j({\mathcal {D}})}}\right] \le b^{-1} \log \sum _{l=1}^{N_D} {\mathbb {E}}\left[ e^{b\Vert G\Vert _{C^j({\mathbb {B}}_l)}}\right] \\\le & {} b^{-1} \log \left( N_{{\mathcal {D}}} {\mathbb {E}}\left[ e^{b\Vert G\Vert _{C^j({\mathbb {B}}_1)}}\right] \right) \\\le & {} b^{-1} \log \left( N_{{\mathcal {D}}} e^{b{\tilde{\sigma }}(c_1 +k\sqrt{2\pi } ) + b^2 {\tilde{\sigma }}^2/2} \right) \qquad \textrm{using } C.5\\= & {} b^{-1}\log (N_{{\mathcal {D}}})+ {\tilde{\sigma }}(c_1 +k\sqrt{2\pi } )+ b\frac{{\tilde{\sigma }}^2}{2} =: h(b). \end{aligned}$$

Differentiating h with respect to b, we find that $h(b)\le h(b_0)$ for $b_0=\sqrt{2}\sqrt{\log ( N_{{\mathcal {D}}}})/{\tilde{\sigma }}$ and thus

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}\right] \le h(b_0) = \sqrt{2}{\tilde{\sigma }}\sqrt{\log (N_{{\mathcal {D}}})}+ {\tilde{\sigma }}(c_1 +k\sqrt{2\pi } ) =: \mu . \end{aligned}$$

(C.6)

Now let $p\ge 1$. Then, using the inequality

$$\begin{aligned} {\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathcal {D}})}> \mu + u \right\} \le ke^{-\frac{u^2}{2{\tilde{\sigma }}^2}}, \quad u>0 \end{aligned}$$

together with (C.6), yields

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}^p \right]= & {} p\int _{0}^{\infty } t^{p-1} {\mathbb {P}}\left\{ \Vert G\Vert _{C^j({\mathcal {D}})} > \mu + (t-\mu ) \right\} dt \\\le & {} \mu ^p + pk\int _{\mu }^{\infty } t^{p-1}e^{-\frac{(t-\mu )^2}{2{\tilde{\sigma }}^2}} dt \le \mu ^p + pk\int _{{\mathbb {R}}} |t|^{p-1}e^{-\frac{(t-\mu )^2}{2{\tilde{\sigma }}^2}} dt\\= & {} \mu ^p + pk \sqrt{2\pi }{\tilde{\sigma }}{\mathbb {E}}\left[ |Z|^{p-1}\right] , \qquad Z\sim {\mathcal {N}}(\mu ,{\tilde{\sigma }}^2). \end{aligned}$$

Now for $Z\sim {\mathcal {N}}(\mu ,{\tilde{\sigma }}^2)$ and $Z':= (Z-\mu )/{\tilde{\sigma }} \sim {\mathcal {N}}(0,1)$,

$$\begin{aligned} {\mathbb {E}}\left[ |Z|^{p-1}\right]{} & {} = {\tilde{\sigma }}^{p-1} {\mathbb {E}}\left[ |Z'+\mu /{\tilde{\sigma }}|^{p-1}\right] \\{} & {} \le 2^{p-2}{\tilde{\sigma }}^{p-1}\left( {\mathbb {E}}\left[ |Z'|^{p-1}\right] + (\mu /{\tilde{\sigma }})^{p-1} \right) \\{} & {} =: C_p({\tilde{\sigma }}^{p-1}+\mu ^{p-1}), \end{aligned}$$

where $C_p:= 2^{p-2} {\mathbb {E}}\left[ |Z'|^{p-1}\right]$ depends only on p, so that

$$\begin{aligned} {\mathbb {E}}\left[ \Vert G\Vert _{C^j({\mathcal {D}})}^p \right] \le \mu ^p + pk \sqrt{2\pi }{\tilde{\sigma }}C_p ({\tilde{\sigma }}^{p-1}+\mu ^{p-1}). \end{aligned}$$

The conclusion follows from the definition of $\mu$ in (C.6) and the fact that there are constants $C_1,C_2>0$ such that $C_1 \textrm{vol}({\mathcal {D}})\le N_{{\mathcal {D}}} \le C_2\textrm{vol}({\mathcal {D}})$. $\square$