Gaussian Random Measures Generated by Berry’s Nodal Sets

Abstract

We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry’s random wave model to a finite collection of (possibly overlapping) smooth compact subsets of ${\mathbb {R}}^2$. Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points.

Ancillary Results from [30] and More

1.1 Covariances

In [30, Lemma 3.1], the authors computed the distribution of the Gaussian vector $(B_E(x), B_E(y), \nabla B_E(x), \nabla B_E(y))\in {\mathbb {R}}^6$ for $x, y\in {\mathbb {R}}^2$, where $\nabla B_E$ is the gradient field $\nabla := (\partial _1, \partial _2), \partial _i := \partial _{x_i} = \partial /\partial {x_i}$ for $i=1,2$. For $i,j\in \lbrace 0,1,2 \rbrace$ define

$$\begin{aligned} r^E_{i,j}(x-y) := \partial _{x_i} \partial _{y_j} r^E(x-y), \end{aligned}$$

(A.1)

with $\partial _{x_0}$ and $\partial _{y_0}$ equal to the identity by definition.

Lemma A.1

([30, Lemma 3.1]) The centered Gaussian vector

$$\begin{aligned} (B_E(x), B_E(y), \nabla B_E(x), \nabla B_E(y))\in {\mathbb {R}}^6, \quad x\ne y\in {\mathbb {R}}^2, \end{aligned}$$

has the following covariance matrix:

$$\begin{aligned} \Sigma ^E (x-y)= \begin{pmatrix} \Sigma _{1}^E(x-y) &{}\Sigma _{2}^E(x-y)\\ \Sigma _{2}^E(x-y)^t &{}\Sigma _{3}^E(x-y) \end{pmatrix}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} \Sigma _1^E(x-y) = \begin{pmatrix} 1 &{}r^E(x-y)\\ r^E(x-y) &{}1 \end{pmatrix}, \end{aligned}$$

$r^E$ being defined in (2.1),

$$\begin{aligned} \Sigma _2^E(x-y) = \begin{pmatrix} 0 &{}0 &{}r_{0,1}^E(x-y) &{}r_{0,2}^E(x-y)\\ -r_{0,1}^E(x-y) &{}-r_{0,2}^E(x-y) &{}0 &{}0 \end{pmatrix}, \end{aligned}$$

(A.3)

with, for $i=1,2$,

$$\begin{aligned} r_{0,i}^E(x-y) = 2\pi \sqrt{E} \,\frac{x_i-y_i}{\Vert x-y\Vert }\, J_1(2\pi \sqrt{E}\Vert x-y\Vert ). \end{aligned}$$

Finally

$$\begin{aligned} \Sigma _3^E(x-y) = \begin{pmatrix} 2\pi ^2E &{}0 &{}r^E_{1,1}(x-y) &{}r^E_{1,2}(x-y)\\ 0 &{}2\pi ^2 E &{}r^E_{2,1}(x-y) &{}r^E_{2,2}(x-y)\\ r^E_{1,1}(x-y) &{}r^E_{2,1}(x-y) &{}2\pi ^2E &{}0\\ r^E_{1,2}(x-y) &{}r^E_{2,2}(x-y) &{}0 &{}2\pi ^2E \end{pmatrix}, \end{aligned}$$

where for $i=1,2$

$$\begin{aligned} r^E_{i,i}(x-y)= 2\pi ^2 E \left( J_0(2\pi \sqrt{E}\Vert x-y\Vert ) + \Big (1 - 2\frac{(x_i - y_i)^2}{\Vert x-y\Vert ^2} \Big ) J_2(2\pi \sqrt{E}\Vert x-y\Vert ) \right) ,\nonumber \\ \end{aligned}$$

(A.4)

and

$$\begin{aligned} r_{12}^E(x-y) = r^E_{2,1}(x-y)= -4\pi ^2 E \frac{(x_1 - y_1)(x_2 - y_2)}{\Vert x - y\Vert ^2} J_2(2\pi \sqrt{E}\Vert x-y\Vert ). \end{aligned}$$

(A.5)

Let us also define, for $k,l\in \lbrace 0,1,2\rbrace$,

$$\begin{aligned} {{\widetilde{r}}}^E_{k,l}(x,y) = {{\widetilde{r}}}^E_{k,l}(x-y) := {\mathbb {E}}\left[ {{\widetilde{\partial }}}_k B_E(x) {{\widetilde{\partial }}}_l B_E(y) \right] ,\qquad x,y\in {\mathbb {R}}^2, \end{aligned}$$

with ${{\widetilde{\partial }}}_0 B_E := B_E$, where we define the normalized derivatives as

$$\begin{aligned} {{\widetilde{\partial }}}_i := \frac{\partial _i}{\sqrt{2\pi ^2E}},\qquad i=1,2, \end{aligned}$$

(A.6)

and accordingly the normalized gradient ${{\widetilde{\nabla }}}$ as

$$\begin{aligned} {{\widetilde{\nabla }}} := ({{\widetilde{\partial }}}_1, {{\widetilde{\partial }}}_2) = \frac{\nabla }{\sqrt{2\pi ^2E}}. \end{aligned}$$

(A.7)

One has the following uniform estimate for Bessel functions: As $\phi \rightarrow \infty$,

(A.8)

uniformly on $(\phi , \theta )$, where the constants involved in the O-notation do not depend on E. As $\psi \rightarrow 0$,

$$\begin{aligned}&r^1(\psi \cos \theta , \psi \sin \theta )\longrightarrow 1, \qquad {{\widetilde{r}}}^1_{0,i}(\psi \cos \theta , \psi \sin \theta )= O(\psi ),\nonumber \\&{{\widetilde{r}}}^1_{i,i}(\psi \cos \theta , \psi \sin \theta )\longrightarrow 1,\qquad {{\widetilde{r}}}^1_{1,2}(\psi \cos \theta , \psi \sin \theta )=O(\psi ^2), \end{aligned}$$

(A.9)

uniformly on $\theta$, for $i=1,2$.

Remark A.1

It is important to stress that the planar random waves can be formally represented as a stochastic integral with respect to a Gaussian random measure W, in the following way

$$\begin{aligned} B_E(x)= \int _0^\pi \, f_E(x,t) \, dW(t)= I_1\left( f_E(x,\cdot )\right) , \end{aligned}$$

(A.10)

where $f_E$ is chosen in such a way that

$$\begin{aligned} {\mathbb {E}}\left[ B_E(x)B_E(y)\right]&= J_0(2\pi \sqrt{E}\left\Vert x-y\right\Vert )\\&= \int _0^\pi \,\cos \left( 2\pi \sqrt{E}\left\Vert x-y\right\Vert \,\sin t\right) \,dt = \int _0^\pi \,f_E(x,t)\,f_E(y,t)\,dt. \end{aligned}$$

1.2 Chaos

We refer the reader to [29, Chapter 2] and [35, Chapter 5] for a self-contained introduction to Wiener chaos. The next result contains an explicit description of the chaotic expansions of ${\mathscr {L}} _E(z):=\text {length} (B_E^{-1}(z)\cap D)$ and ${\mathscr {N}}_E(z) := \#\left( \left( B_E^{\mathbb {C}}\right) ^{-1}(z)\cap D\right)$, $z\in {\mathbb {R}}$.

Proposition A.2

The chaotic expansion of the level curve length in D is

$$\begin{aligned} \begin{aligned} {\mathscr {L}} _E(z) = \sum _{q=0}^{+\infty } {\mathscr {L}} _E^{[q]}(z) =&\sqrt{2\pi ^2E} \sum _{q=0}^{+\infty } \sum _{u=0}^{q} \sum _{m=0}^{u} \beta _{q - u}(z) \alpha _{m, u - m} \\&\times \int _{D} H_{q-u}(B_E(x)) H_{m}({{\widetilde{\partial }}}_1 B_E(x)) H_{u-m}({{\widetilde{\partial }}}_2 B_E(x))\,dx, \end{aligned} \end{aligned}$$

(A.11)

where $\lbrace \beta _{n}(z)\rbrace _{n\ge 0}$ are the formal coefficients of the chaotic expansion of $\delta _z$ (see Remark A.2), while $\lbrace \alpha _{n,m}\rbrace _{n,m\ge 0}$ is the sequence of chaotic coefficients of the Euclidean norm in ${\mathbb {R}}^2$$\Vert \cdot \Vert$ appearing in [26, Lemma 3.5]. Here, the symbol ${\mathscr {L}} _E^{[q]}(z)$ indicates the projection of ${\mathscr {L}} _E(z)$ onto the qth Wiener chaos associated with $B_E$, as defined in [29, Section 2.2].

For the number of level points in D we have

$$\begin{aligned}&{\mathscr {N}}_E(z) = \sum _{q=0}^{+\infty } {\mathscr {N}}_E^{[q]}(z) = 2\pi ^2E \sum _{q=0}^{+\infty } \sum _{i_1+i_2+i_3+j_1+j_2+j_3=q} \beta _{i_1}(z) \beta _{j_1}(z)\,\zeta _{i_2,i_3,j_2,j_3}\nonumber \\&\quad \int _{D} H_{i_1}(B_E(x)) H_{j_1}({{\widehat{B}}}_E(x)) H_{i_2}({{\widetilde{\partial }}}_1 B_E(x)) H_{i_3}({{\widetilde{\partial }}}_2 B_E(x))H_{j_2}({{\widetilde{\partial }}}_1 {{\widehat{B}}}_E(x)) H_{j_3}({{\widetilde{\partial }}}_2 {{\widehat{B}}}_E(x))\,dx,\nonumber \\ \end{aligned}$$

(A.12)

where $i_2, i_3, j_2, j_3$ have the same parity; here the sequence $\lbrace \zeta _{i_2,i_3,j_2,j_3}\rbrace$ corresponds to the chaotic expansion of the absolute value of the Jacobian appearing in [19, Lemma 4.2]. Here, the symbol ${\mathscr {N}}_E^{[q]}(z)$ indicates the projection of ${\mathscr {N}}_E(z)$ onto the qth Wiener chaos associated with $B^{{\mathbb {C}}}_E$, as defined in [29, Section 2.2].

Remark A.2

The coefficients $\beta _l$ are defined as the limit, as $\varepsilon \rightarrow 0$, of $\beta ^\varepsilon _l:=\frac{1}{l!}\eta _l^\varepsilon (z)$, where

$$\begin{aligned} \frac{1}{2\varepsilon }\mathbb {1}_{[z-\varepsilon ,z+\varepsilon ]}(\cdot )=\sum _{l=0}^\infty \,\frac{1}{l!}\,\eta _l^\varepsilon (z)\,H_l(\cdot ). \end{aligned}$$

In [36, Proposition 7.2.2], it is shown that

$$\begin{aligned} \eta _{n}(z)&=\lim _{\varepsilon \longrightarrow 0} \frac{1}{2\varepsilon } \int _{z-\varepsilon }^{z+\varepsilon } \gamma (t) H_{n}(t) \,dt=\lim _{\varepsilon \longrightarrow 0} \frac{1}{2\varepsilon } \int _{z-\varepsilon }^{z+\varepsilon } \gamma (t) (-1)^{n} \gamma ^{-1}(t) \frac{d^{n}}{dt^{n}} \gamma (t) \,dt \nonumber \\&=\lim _{\varepsilon \longrightarrow 0}\frac{(-1)^n}{2\varepsilon } \int _{z-\varepsilon }^{z+\varepsilon } \frac{d^{n}}{dt^{n}} \gamma (t) \,dt =\gamma (z)\,H_{n}(z). \end{aligned}$$

(A.13)

with $\gamma$ the standard Gaussian density on ${\mathbb {R}}$ and

$$\begin{aligned} \alpha _{n,n-m}=\frac{1}{2\pi \, (n)!\,(n-m)!} \int _{{\mathbb {R}}^2} \sqrt{y^2 + z^2} \, H_{n}(y) H_{n-m}(z) {\mathrm {e}}^{-\frac{y^2+z^2}{2}}\,dy dz, \end{aligned}$$

(A.14)

where (A.14) vanishes whenever n or $n-m$ is odd. In [19], it is shown that

$$\begin{aligned} \zeta _{a,b,c,d}=\frac{1}{a!\,b!\,c!\,d!\,}\,\,{\mathbb {E}}\left[ \left|XY-ZW\right|\,H_a(X)H_b(Y)H_c(Z)H_d(W)\right] , \end{aligned}$$

where (X, Y, V, W) is a standard real four-dimensional Gaussian vector.

In particular, we have

$$\begin{aligned}&\beta _0(z)=\gamma (z)H_0(z)=\gamma (z),\quad \beta _1(z)=\gamma (z)H_1(z)=\gamma (z)\,z, \nonumber \\&\beta _2(z)=\frac{1}{2}\,\gamma (z)H_2(z)=\frac{1}{2}\,\gamma (z)(z^2-1), \quad \beta _3(z)= \frac{1}{6}\,\gamma (z)H_3(z)=\frac{1}{6}\,\gamma (z)(z^3-3z), \nonumber \\&\beta _4=\frac{1}{24}\,\gamma (z)H_4(z)=\frac{1}{24}\,\gamma (z)(z^4-6z^2+3), \end{aligned}$$

(A.15)

$$\begin{aligned}&\alpha _{0,0}=\frac{\sqrt{2\pi }}{2},\quad \alpha _{2,0}=\alpha _{0,2}=\frac{\sqrt{2\pi }}{8},\quad \alpha _{4,0}=\alpha _{0,4}=-\frac{\sqrt{2\pi }}{128},\quad \alpha _{2,2}=-\frac{\sqrt{2\pi }}{64} \end{aligned}$$

(A.16)

and

$$\begin{aligned} \begin{aligned} \zeta _{0,0,0,0}&=1,\quad \zeta _{2,0,0,0}=\zeta _{0,2,0,0}=\zeta _{0,0,2,0}=\zeta _{0,0,0,2}=\frac{1}{4},\\ \zeta _{1,1,1,1}&=-\frac{3}{8},\quad \zeta _{2,2,0,0}=\zeta _{0,0,2,2}=-\frac{1}{32},\\ \zeta _{2,0,2,0}&=\zeta _{0,2,0,2}=-\frac{1}{32},\quad \zeta _{2,0,0,2}=\zeta _{0,2,2,0}=\frac{5}{32},\\ \zeta _{4,0,0,0}&=\zeta _{0,4,0,0}=\zeta _{0,0,4,0}=\zeta _{0,0,0,4}=-\frac{3}{192}. \end{aligned} \end{aligned}$$

(A.17)

Note that, when $z=0$, the odd-chaoses vanish.

Once the chaotic expansions were established, the authors of [30] proved that, as $E\rightarrow +\infty$ (see [30, Equation (2.29)])

$$\begin{aligned} \frac{{\mathscr {L}} _E - {\mathbb {E}}[{\mathscr {L}} _E]}{\sqrt{{{\,\mathrm{Var}\,}}({\mathscr {L}} _E)}} = \frac{{\mathscr {L}} _E^{[4]}}{\sqrt{{{\,\mathrm{Var}\,}}({\mathscr {L}} _E^{[4]})}} + o_{{\mathbb {P}}}(1), \qquad \frac{{\mathscr {N}}_E - {\mathbb {E}}[{\mathscr {N}}_E]}{\sqrt{{{\,\mathrm{Var}\,}}({\mathscr {N}}_E)}} = \frac{{\mathscr {N}}_E^{[4]}}{\sqrt{{{\,\mathrm{Var}\,}}({\mathscr {N}}_E^{[4]})}} + o_{{\mathbb {P}}}(1) \end{aligned}$$

using the following results (and in particular that ${{\,\mathrm{Var}\,}}{\mathscr {L}} _E \sim {{\,\mathrm{Var}\,}}{\mathscr {L}} _E^{[4]}$).

Lemma A.3

[30, Lemma 4.1 and 4.2] We have

$$\begin{aligned} {\mathscr {L}} _E^{[2]} = \frac{1}{8\pi \sqrt{2\,E}} \int _{\partial D} B_E(x)\langle \nabla B_E(x),n(x)\rangle dx, \end{aligned}$$

(A.18)

where n(x) is the outward pointing normal at x, and hence

$$\begin{aligned} {{\,\mathrm{Var}\,}}({\mathscr {L}} _E^{[2]}) = O(1). \end{aligned}$$

(A.19)

Moreover,

$$\begin{aligned} {\mathscr {N}}_E^{[2]} = \sqrt{2E}\big ({\mathscr {L}} _E^{[2]}+{\widetilde{{\mathscr {L}} }}_E[2]\big ) \end{aligned}$$

(A.20)

and hence

$$\begin{aligned} {{\,\mathrm{Var}\,}}({\mathscr {N}}_E^{[2]}) = O(E). \end{aligned}$$

(A.21)

Proposition A.4

[30, Proposition 6.1] The fourth chaotic component of ${\mathscr {L}} _E$ is given by

$$\begin{aligned} {\mathscr {L}} _E^{[4]}(D)=\frac{\sqrt{2\pi ^2\,E}}{128}\left\{ 8\,a_{1,E}-a_{2,E}-a_{3,E}-2\,a_{4,E}-8\,a_{5,E}-8\,a_{6,E}\right\} , \end{aligned}$$

(A.22)

where

$$\begin{aligned} \begin{aligned} a_{1,E}&:=\int _{D} H_4(B_E(x))dx,\quad a_{2,E}:=\int _{D} H_4({\widetilde{\partial }}_1 B_E(x))dx,\quad a_{3,E}:=\int _{D} H_4({\widetilde{\partial }}_2 B_E(x))dx,\\ a_{4,E}&:= \int _{D} H_2({\widetilde{\partial }}_1 B_E(x)) H_2({\widetilde{\partial }}_2 B_E(x))dx,\\ a_{5,E}&:= \int _{D} H_2(B_E(x)) H_2({\widetilde{\partial }}_1 B_E(x))dx,\quad a_{6,E} := \int _{D} H_2(B_E(x))H_2({\widetilde{\partial }}_2 B_E(x))dx. \end{aligned} \end{aligned}$$

(A.23)

Its variance satisfies

$$\begin{aligned} \begin{aligned} {{{\,\mathrm{Var}\,}}}({\mathscr {L}} _E^{[4]})&= \frac{\pi ^2E}{8192}\,{{{\,\mathrm{Var}\,}}}\left( 8a_{1,E}-a_{2,E}-a_{3,E}-2a_{4,E}-8a_{5,E}-8a_{6,E} \right) \\&\sim \frac{{{\text {area}}}(D)\,\log E}{512\pi }, \end{aligned} \end{aligned}$$

(A.24)

where the last asymptotic equivalence holds as $E\rightarrow +\infty$.

Proposition A.5

[30, Proposition 6.2] The fourth chaotic component of ${\mathscr {N}}_E$ is given by

$$\begin{aligned} {\mathscr {N}}_E^{[4]}(D)=a_{E}(D)+{\widehat{a}}_{E}(D)+b_{E}(D), \end{aligned}$$

(A.25)

where

$$\begin{aligned} a_E(D)=\frac{\pi \,E}{64}\left\{ 8\,a_{1,E}(D)-a_{2,E}(D)-2a_{3,E}(D)-8\,a_{4,E}(D)\right\} , \end{aligned}$$

${\widehat{a}}_{E}(D)$ is defined in the same way as $a_{E}(D)$, except for the fact that one uses ${\widehat{B}}_E$ instead of $B_E$, and

$$\begin{aligned} b_E&= \frac{\pi E}{8}\Big \lbrace 2b_{1,E} - b_{2,E} - b_{3,E} -b_{4,E} - b_{5,E} - \frac{1}{4} b_{6,E} -\frac{1}{4} b_{7,E} \\&\quad + \frac{5}{4} b_{8,E} + \frac{5}{4} b_{9,E}- 3b_{10,E} \Big \rbrace , \end{aligned}$$

with $a_{i,E}$, $i=1,\ldots ,4$ defined in (A.23) and

$$\begin{aligned} \begin{aligned}&b_{1,E} := \int _{D} H_2(B_E(x))H_2({\widehat{B}}_E(x)) dx \qquad b_{2,E}:=\int _{D} H_2(B_E(x)) H_2({\widetilde{\partial }}_1 {\widehat{B}}_E(x)dx\\&b_{3,E}=\int _{D} H_2(B_E(x)) H_2({\widetilde{\partial }}_2 {\widehat{B}}_E(x))dx\qquad b_{4,E}=\int _{D} H_2({\widetilde{\partial }}_1 B_E(x))H_2({\widehat{B}}_E(x)) dx\\&b_{5,E}:=\int _{D} H_2({\widetilde{\partial }}_2 B_E(x))H_2({\widehat{B}}_E(x)) dx\qquad b_{6,E} := \int _{D} H_2({\widetilde{\partial }}_1 B_E(x))H_2({\widetilde{\partial }}_1 {\widehat{B}}_E(x)) dx\\&b_{7,E} := \int _{D} H_2({\widetilde{\partial }}_2 B_E(x))H_2({\widetilde{\partial }}_2 {\widehat{B}}_E(x)) dx\qquad b_{8,E} := \int _{D} H_2({\widetilde{\partial }}_1 B_E(x))H_2({\widetilde{\partial }}_2 {\widehat{B}}_E(x))dx\\&b_{9,E} := \int _{D} H_2({\widetilde{\partial }}_2 B_E(x))H_2({\widetilde{\partial }}_1 {\widehat{B}}_E(x))dx\\&b_{10,E} := \int _{D} {\widetilde{\partial }}_1 B_E(x){\widetilde{\partial }}_2 B_E(x) {\widetilde{\partial }}_1 {\widehat{B}}_E(x){\widetilde{\partial }}_2 {\widehat{B}}_E(x)dx. \end{aligned} \end{aligned}$$

(A.26)

Its variance satisfies

$$\begin{aligned} {{{\,\mathrm{Var}\,}}}({\mathscr {N}}_E^{[4]})=2{{{\,\mathrm{Var}\,}}}(a_E)+{{{\,\mathrm{Var}\,}}}(b_E)\sim \frac{11{{\text {area}}}(D)}{32\pi }\,E\log E, \end{aligned}$$

(A.27)

where the last asymptotic equivalence holds as $E\rightarrow +\infty$.