Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Gaussian Random Measures Generated by Berry’s Nodal Sets


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.

This is a preview of subscription content, log in to check access.


  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].


  1. 1.

    Angst, J., Pham, V.H., Poly, G.: Universality of the nodal length of bivariate random trigonometric polynomials. Trans. Am. Math. Soc. 370, 8331–8357 (2018)

  2. 2.

    Beliaev, D., Cammarota, V., Wigman, I.: Two point function for critical points of a random plane wave. Int. Math. Res. Not. 2019, 2661–2689 (2017)

  3. 3.

    Berry, M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083–2092 (1977)

  4. 4.

    Berry, M.V.: Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A 35(13), 3025–3038 (2002)

  5. 5.

    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

  6. 6.

    Bogachev, V.I., Miftakhov, A.F.: On weak convergence of finite-dimensional and infinite-dimensional distributions of random processes. Theory Stoch. Process. 21(37), 1–11 (2016)

  7. 7.

    Benatar, J., Maffucci, R.W.: Random waves on \({\mathbb{t}}^3\): nodal area variance and lattice point correlations. Int. Math. Res. Not. 2019, 3032–3075 (2017)

  8. 8.

    Benatar, J., Marinucci, D., Wigman, I.: Planck-scale distribution of nodal length of arithmetic random waves. J. Anal. Math. (in press)

  9. 9.

    Bass, R.F., Pyke, R.: The space \(D(A)\) and weak convergence for set-indexed processes. Ann. Probab. 13(3), 860–884 (1985)

  10. 10.

    Beliaev, D., Wigman, I.: Volume distribution of nodal domains of random band-limited functions. Probab. Theory Relat. Fields 172(1), 453–492 (2018)

  11. 11.

    Cammarota, V.: Nodal area distribution for arithmetic random waves. Trans. Am. Math. Soc. 372, 3539–3564 (2019)

  12. 12.

    Canzani, H., Hanin, B.: Local universality for zeros and critical points of monochromatic random waves. arXiv:610.09438 (2016)

  13. 13.

    Cammarota, V., Marinucci, D.: A quantitative central limit theorem for the Euler–Poincarè characteristic of random spherical eigenfunctions. Ann. Probab. 46(6), 3188–3228 (2018)

  14. 14.

    Cammarota, V., Marinucci, D.: On the correlation of critical points and angular trispectrum for random spherical harmonics. arXiv:1907.05810 (2019)

  15. 15.

    Cammarota, V., Marinucci, D.: A reduction principle for the critical values of random spherical harmonics. Stoch. Process. Appl. (2019).

  16. 16.

    Cammarota, V., Marinucci, D., Wigman, I.: Fluctuations of the Euler–Poincaré characteristic for random spherical harmonics. Proc. Am. Math. Soc. 144, 4759–4775 (2016)

  17. 17.

    Cammarota, V., Marinucci, D., Wigman, I.: On the distribution of the critical values of random spherical harmonics. J. Geom. Anal. 26, 3252–3324 (2016)

  18. 18.

    Canzani, H., Sarnak, P.: Topology and nesting of the zero set components of monochromatic random waves. Commun. Pure Appl. Math. 72(2), 343–374 (2019)

  19. 19.

    Dalmao, F., Nourdin, I., Peccati, G., Rossi, M.: Phase singularities in complex arithmetic random waves. Electron. J. Probab. 24, 45 (2019)

  20. 20.

    Dennis, M.R., O’Holleran, K., Padgett, M.J.: Singular optics: optical vortices and polarization singularities. Prog. Opt. 53, 293–363 (2016)

  21. 21.

    Kac, M.: On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Am. Math. Soc. 59, 401–414 (1946)

  22. 22.

    Keeler, Blake: A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points. arXiv:1905.05136 (2019)

  23. 23.

    Krishnapur, M., Kurlberg, P., Wigman, I.: Non-universality of nodal length distribution for arithmetic random waves. Ann. Math. 177(2), 699–737 (2013)

  24. 24.

    Kurlberg, P., Wigman, I.: Non-universality of the Nazarov–Sodin constant for random plane waves and arithmetic random waves. Adv. Math. 330, 516–552 (2018)

  25. 25.

    Marinucci, D., Peccati, G.: Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2011)

  26. 26.

    Marinucci, D., Peccati, G., Rossi, M., Wigman, I.: Non-universality of nodal length distribution for arithmetic random waves. GAFA 3, 926–960 (2016)

  27. 27.

    Marinucci, D., Rossi, M., Wigman, I.: The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics. Ann. Inst. Henri Poincaré Probab. Stat. (in press)

  28. 28.

    Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat. 42(4), 1285–1295 (1971)

  29. 29.

    Nourdin, I., Peccati, G.: Normal Approximation with Malliavin Calculus: From Stein’s Method to Universality. Cambridge University Press, Cambridge (2012)

  30. 30.

    Nourdin, I., Peccati, G., Rossi, M.: Nodal statistics of planar random waves. Commun. Math. Phys. 369(1), 99–151 (2019)

  31. 31.

    Nazarov, F., Sodin, M.: Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions. J. Math. Phys. Anal. Geom. 12(3), 205–278 (2016)

  32. 32.

    Oravecz, F., Rudnick, D., Wigman, I.: The Leray measure of nodal sets for random eigenfunctions on the torus. Ann. Inst. Fourier 58(1), 299–335 (2008)

  33. 33.

    Peccati, G., Rossi, M.: Quantitative Limit Theorems for Local Functionals of Arithmetic Random Waves. Abel Symposium 2016 (Springer) (2018)

  34. 34.

    Peccati, G., Tudor, C.A.: Gaussian limits for vector-valued multiple stochastic integrals. Sémin. Probab. 38, 247–262 (2005)

  35. 35.

    Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagrams. Springer-Verlag, Berlin (2010)

  36. 36.

    Rossi, M.: On the High Energy Behavior of Nonlinear Functionals of Random Eigenfunctions on \({\mathbb{S}}^d\). In: 19th European Young Statisticians Meeting in Prague, pp. 119–124 (2015)

  37. 37.

    Rusakov, A., Seleznjev, O.: On weak convergence of functionals on smooth random functions. Math. Commun. 6, 123–134 (2001)

  38. 38.

    Rudnick, Z., Wigman, I.: On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9(1), 109–130 (2008)

  39. 39.

    Rossi, M., Wigman, I.: Asymptotic distribution of nodal intersections for arithmetic random waves. Nonlinearity 31(10), 4472 (2018)

  40. 40.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin (1999)

  41. 41.

    Sarnak, P., Wigman, I.: Topologies of nodal sets of random band limited functions. In: Advances in the Theory of Automorphic Forms and Their \(L\)-functions, Volume 664 of Contemporary Mathematics (2016)

  42. 42.

    Takacs, L.: On the distribution of the integral of the absolute value of the brownian motion. Ann. Appl. Probab. 3(1), 186–197 (1993)

  43. 43.

    Todino, A.P.: Nodal lengths in shrinking domains for random eigenfunctions on \({\mathbb{S}}^2\). arXiv:1807.11787 (2018)

  44. 44.

    Todino, A.P.: A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of \({\mathbb{S}}^2\). J. Math. Phys. 60(2), 023505 (2019)

  45. 45.

    Urbina, J., Richter, K.: Random quantum states: recent developments and applications. Adv. Phys. 62, 787–831 (2013)

  46. 46.

    Wigman, I.: Fluctuations of the nodal length of random spherical harmonics. Commun. Math. Phys. 298(3), 787–831 (2010)

  47. 47.

    Zelditch, S.: Real and complex zeros of Riemannian random waves. In: Spectral Analysis in Geometry and Number Theory, Volume 484 of Contemporary Mathematics (2009)

Download references


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.

Author information

Correspondence to Anna Vidotto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Eric A. Carlen.

Ancillary Results from [30] and More

Ancillary Results from [30] and More


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

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

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:

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


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

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

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

with, for \(i=1,2\),

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


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

where for \(i=1,2\)

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


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

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

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

and accordingly the normalized gradient \({{\widetilde{\nabla }}}\) as

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

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


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

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

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

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

where \(f_E\) is chosen in such a way that

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


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

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

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

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

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

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

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

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

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

$$\begin{aligned}&\beta _0(z)=\gamma (z)H_0(z)=\gamma (z),\quad \beta _1(z)=\gamma (z)H_1(z)=\gamma (z)\,z, \nonumber \\&\beta _2(z)=\frac{1}{2}\,\gamma (z)H_2(z)=\frac{1}{2}\,\gamma (z)(z^2-1), \quad \beta _3(z)= \frac{1}{6}\,\gamma (z)H_3(z)=\frac{1}{6}\,\gamma (z)(z^3-3z), \nonumber \\&\beta _4=\frac{1}{24}\,\gamma (z)H_4(z)=\frac{1}{24}\,\gamma (z)(z^4-6z^2+3), \end{aligned}$$
$$\begin{aligned}&\alpha _{0,0}=\frac{\sqrt{2\pi }}{2},\quad \alpha _{2,0}=\alpha _{0,2}=\frac{\sqrt{2\pi }}{8},\quad \alpha _{4,0}=\alpha _{0,4}=-\frac{\sqrt{2\pi }}{128},\quad \alpha _{2,2}=-\frac{\sqrt{2\pi }}{64} \end{aligned}$$


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

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

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

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

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


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

and hence

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

Proposition A.4

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

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


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

Its variance satisfies

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

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

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


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

\({\widehat{a}}_{E}(D)\) 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

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

Its variance satisfies

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

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Peccati, G., Vidotto, A. Gaussian Random Measures Generated by Berry’s Nodal Sets. J Stat Phys 178, 996–1027 (2020).

Download citation


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

Mathematics Subject Classification

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