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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.

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Notes

  1. 1.

    Note that, if \({\text {diam}}(D_1 \cap D_2)\le \delta \), there is no need of splitting the integral in the sum of \(C_{2,2,1}\) and \(C_{2,2,2}\), as in this case

    $$\begin{aligned} \int _{1}^{\sqrt{E}{\text {diam}}(D_1 \cap D_2)}\,\frac{d\psi }{\psi }{\text {area}}\left( D_1\cap \left( D_2^{+\frac{\psi }{\sqrt{E}}}\setminus D_2^{-\frac{\psi }{\sqrt{E}}}\right) \right) \le \int _{1}^{\delta \sqrt{E}}\frac{d\psi }{\psi }{\text {area}}\left( D_1\cap \left( D_2^{+\frac{\psi }{\sqrt{E}} }\setminus D_2^{-\frac{\psi }{\sqrt{E}}}\right) \right) , \end{aligned}$$

    and the last integral equals \(C_{2,2,1}\).

  2. 2.

    Note that, if \({\text {diam}}(D_1 \cup D_2)\le \delta \), there is no need of splitting the integral in the sum of \(I_1\) and \(I_2\), as

    $$\begin{aligned} \int _{1}^{\sqrt{E} {\text {diam}}(D_1 \cup D_2)}\,\frac{1}{\psi }\,d\psi \,{\text {area}}\left( D_1\cap D_2^{+\psi /\sqrt{E}}\right) \le \int _{1}^{\delta \sqrt{E}}\,\frac{1}{\psi }\,d\psi \,{\text {area}}\left( D_1\cap D_2^{+\psi /\sqrt{E}}\right) =I_1. \end{aligned}$$
  3. 3.

    Note that, if \(\delta \) is such that \({\text {diam}}(D_1 \cup D_2)\le \delta \), there is no need of splitting the integral in the sum of \(C_{1,1}\) and \(C_{2,2}\), as

    $$\begin{aligned} \int _{1}^{\sqrt{E}{\text {diam}}(D_1 \cap D_2)}\,\frac{1}{\psi }\,{\text {area}}\left( D_1\cap D_2^{-\psi /\sqrt{E}}\right) \,d\psi \le \int _{1}^{\delta \sqrt{E}}\,\frac{1}{\psi }\,{\text {area}}\left( D_1\cap D_2^{-\psi /\sqrt{E}}\right) \,d\psi = C_{1,1} \end{aligned}$$
  4. 4.

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

  5. 5.

    see also [40, Theorem 2.1].

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Acknowledgements

The authors would like to thank Maurizia Rossi for insightful discussions. AV would like also to thank Guangqu Zheng for some useful comments on an early version of this work. Part of this work has been written in the framework of AFR research project High-dimensional challenges and non-polynomial transformations in probabilistic approximations (HIGH-NPOL) funded by FNR—Luxembourg National Research Fund. Giovanni Peccati is also supported by the FNR Grant FoRGES (R-AGR-3376-10) at Luxembourg University. Anna Vidotto acknowledges also the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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Correspondence to Anna Vidotto.

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Communicated by Eric A. Carlen.

Ancillary Results from [30] and More

Ancillary Results from [30] and More

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

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

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 (XYVW) 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 \(E\rightarrow +\infty \).

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Peccati, G., Vidotto, A. Gaussian Random Measures Generated by Berry’s Nodal Sets. J Stat Phys 178, 996–1027 (2020). https://doi.org/10.1007/s10955-019-02477-z

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Keywords

  • Random plane waves
  • Gaussian random measures
  • Weak convergence
  • Wiener sheet
  • Bessel functions

Mathematics Subject Classification

  • 60G60
  • 60F05
  • 34L20
  • 33C10