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Scars in random waves and the FGF 1/2 universality class

T0 review · 2 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Observables in Berry's random waves whose fluctuations fully correlate with their second Wiener chaos projection converge to the fractional Gaussian field with Hurst index (1-d)/2 that also governs the Poisson line process.

desk verdict Paper ties random-wave observables to FGF 1/2 class via second-chaos correlation and Poisson-line approximation, but the correlation step is the part that needs referee scrutiny. read the letter →

arxiv 2606.07842 v2 pith:NWVE4C6F submitted 2026-06-05 math.PR

Louis Gass , Giovanni Peccati , Michele Stecconi This is my paper · ORCID
classification math.PR
keywords BerryrandomwavesfractionalGaussianfieldPoissonlineprocessWienerchaosuniversalityclassscarslargedomainasymptoticsstationaryfields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that geometric observables in Berry's random wave model on R^d fall into a shared universality class whenever their fluctuations are asymptotically fully correlated with the second Wiener chaos projection. This class is governed by a fractional Gaussian field of Hurst index H=(1-d)/2 and also contains the stationary Poisson line process in R^d. The result implies that raw observables such as critical point counts or level set volumes in the waves have large-domain fluctuations that approximate those of a possibly noisy Poisson line process in the sense of random tempered distributions. This supplies a probabilistic account of filamentary patterns called scars. The work further identifies the scaling limit of quadratic transformations of the associated Radon-Fourier coefficients and gives conditions for the same limits on compact Riemannian manifolds.

What carries the argument

The asymptotic full correlation of an observable's fluctuations with its second Wiener chaos projection, which places the observable in the fractional Gaussian field universality class with Hurst index H=(1-d)/2 shared with the Poisson line process.

What would settle it

A computation or simulation in which the large-domain covariance of critical point counts or level set volumes in Berry waves deviates from the covariance of the fractional Gaussian field with H=(1-d)/2 would falsify the claimed convergence.

Watch

Extended reading notes

Core claim

Any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index H=(1-d)/2; this class includes the classical stationary Poisson line process in R^d. Suitable raw observables of Berry's random wave have large-domain fluctuations that become arbitrarily close, in the sense of random tempered distributions, to those generated by a possibly noisy Poisson line process.

Load-bearing premise

The fluctuations of the observable are asymptotically fully correlated with its second Wiener chaos projection.

Editorial extensions

If this is right

  • Critical point counts and non-nodal level set volumes in random waves have large-domain fluctuations that approach those of a possibly noisy Poisson line process.
  • Quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the same fractional Gaussian class under explicit conditions.
  • The scaling limit of quadratic transformations of Radon-Fourier coefficients for random waves is a generalized random field obtained by composing white noise on the affine Grassmannian with a dimension-dependent deterministic operator.
  • Scars observed in numerical simulations of random waves admit a probabilistic interpretation as approximations to Poisson line patterns.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same correlation condition may identify additional observables that fall into this universality class beyond those already checked.
  • Large-scale statistics of random waves could be simulated more efficiently by sampling from the approximating Poisson line process rather than the full wave model.
  • Analogous chaos-projection criteria might classify scaling limits for other stationary random fields with singular spectral measures.
  • The manifold application suggests the universality class persists under pullback to curved geometries when the correlation condition holds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper claims that any observable in Berry's random wave model on R^d whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a universality class governed by the fractional Gaussian field with Hurst index H=(1-d)/2; this class also includes the stationary Poisson line process. Suitable raw observables of random waves (e.g., critical point counts, non-nodal level-set volumes) are asserted to have large-domain fluctuations arbitrarily close in the sense of random tempered distributions to those of a (possibly noisy) Poisson line process, offering a probabilistic interpretation of observed 'scars'. A second part characterizes the scaling limit of quadratic transformations of Radon-Fourier coefficients for a class of stationary fields, showing that random waves yield a generalized random field obtained by composing white noise on the affine Grassmannian with a deterministic operator. Applications to quadratic functionals of pullback monochromatic waves on compact manifolds are mentioned.

Significance. If the results hold, the work identifies a new universality class for geometric observables of fields with singular (sphere-supported) spectral measures and supplies a concrete probabilistic mechanism linking random-wave geometry to Poisson line processes. The explicit characterization of the quadratic Radon-Fourier scaling limit and the distributional approximation to the Poisson process would be substantive contributions to the study of Gaussian fields and random waves.

major comments (2)
  1. [Abstract, first paragraph] Abstract, first paragraph: the reduction of Berry random-wave observables (critical-point counts, non-nodal level-set volumes) to the FGF H=(1-d)/2 class is conditional on the unverified claim that their fluctuations become asymptotically fully correlated with the second Wiener-chaos projection. The abstract asserts this follows from the spectral measure being supported on the sphere, but supplies neither a quantitative rate nor an explicit verification for these geometric functionals; without that step the distributional closeness to the Poisson line process does not follow.
  2. [Abstract] Abstract: the manuscript states precise theorems on scaling limits and universality but, as presented, contains no proofs, error bounds, or verification steps for the central correlation condition or the tempered-distribution approximation. This prevents assessment of derivation gaps in the claimed limits.
minor comments (1)
  1. [Abstract] The reference to Heller, O'Connor and Gehlen (1987) for the term 'scarlets' should be checked for accuracy and completeness in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, clarifying the scope of our results and indicating revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract, first paragraph] Abstract, first paragraph: the reduction of Berry random-wave observables (critical-point counts, non-nodal level-set volumes) to the FGF H=(1-d)/2 class is conditional on the unverified claim that their fluctuations become asymptotically fully correlated with the second Wiener-chaos projection. The abstract asserts this follows from the spectral measure being supported on the sphere, but supplies neither a quantitative rate nor an explicit verification for these geometric functionals; without that step the distributional closeness to the Poisson line process does not follow.

    Authors: The main theorem establishes the FGF universality class conditionally on the stated asymptotic correlation with the second Wiener chaos projection; this is explicit in the manuscript. The abstract then asserts that suitable geometric observables of random waves satisfy the condition (and hence belong to the class) because their spectral measure is sphere-supported. The full text supplies a heuristic argument based on the concentration of the spectrum on the sphere, which forces higher-order chaos terms to vanish in the large-domain limit while the second-chaos projection survives. We acknowledge, however, that no quantitative rate of correlation or explicit verification is given for the concrete examples (critical-point counts, level-set volumes). We will revise by adding a new subsection that derives the correlation condition for these functionals from the sphere support, including a sketch of the error estimate that controls the contribution of higher chaoses. revision: yes

  2. Referee: [Abstract] Abstract: the manuscript states precise theorems on scaling limits and universality but, as presented, contains no proofs, error bounds, or verification steps for the central correlation condition or the tempered-distribution approximation. This prevents assessment of derivation gaps in the claimed limits.

    Authors: The theorems on the FGF scaling limit (conditional on the correlation assumption) and on the quadratic Radon–Fourier scaling limit are stated and proved in the body of the manuscript; the abstract is only a summary. The proofs of the distributional approximation to the Poisson line process rely on the correlation condition together with the explicit characterization of the quadratic limit as white noise on the affine Grassmannian composed with a deterministic operator. If the reviewed version appeared to lack these elements, it may reflect a submission formatting issue. In the revision we will (i) ensure every theorem is followed immediately by its proof or a clear reference to the relevant section, (ii) insert the error-bound sketch for the correlation condition mentioned above, and (iii) add a short appendix containing the tempered-distribution approximation argument with explicit constants where available. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; claims grounded in external Gaussian field properties

full rationale

The paper's central claims characterize a universality class for observables whose fluctuations are asymptotically fully correlated with their second Wiener chaos projection, contrasting with absolutely continuous spectra. This premise is stated explicitly in the abstract as the scope restriction and is tied to the spectral measure support on the sphere (an external property of the random wave model). The scaling limits and FGF H=(1-d)/2 membership are derived from this condition plus known facts about Poisson line processes and Radon-Fourier coefficients, without reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The second part on quadratic transformations of Radon-Fourier coefficients is presented as a characterization result, not a prediction forced by the paper's own inputs. No quoted equations exhibit the patterns of fitted-input-called-prediction or ansatz-smuggled-via-citation.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claims rest on standard properties of stationary Gaussian fields, Wiener chaos decompositions, and the geometry of the affine Grassmannian; no free parameters, ad-hoc entities, or non-standard axioms are introduced in the abstract.

assumptions (2)
  • domain assumption Stationary random fields admit a Wiener chaos expansion whose second-order term governs the asymptotic fluctuations of the chosen geometric observables
    Invoked when the authors restrict to observables 'whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection'.
  • domain assumption The spectral measure of Berry's random wave model satisfies the conditions needed for the Radon-Fourier coefficients to produce the stated scaling limit
    Underlying the characterization of random waves via white noise on the affine Grassmannian.

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Cite this review

Pith. "Pith review of Scars in random waves and the FGF 1/2 universality class." pith.science (2026). https://pith.science/paper/NWVE4C6F

@misc{pith2026260607842,
  author       = {Pith},
  title        = {Pith review of: Scars in random waves and the FGF 1/2 universality class},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWVE4C6F}},
  note         = {Machine review of arXiv:2606.07842}
}
abstract

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=(1-d)/2$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. Our findings show that suitable raw observables of Berry's random wave (such as critical point counts or non-nodal level set volumes) have large-domain fluctuations that become arbitrarily close -- in the sense of random tempered distributions -- to those generated by a (possibly noisy) Poisson line process. This probabilistic approximation provides evidence that the large-scale filamentary patterns observed in numerical simulations of random waves -- often referred to as "scars" or "scarlets" following the numerical investigations of Heller, O'Connor and Gehlen (1987)-- may admit a natural probabilistic interpretation. In the second part of our work, we characterize the scaling limit -- in a distributional sense -- of suitable quadratic transformations of the Radon--Fourier coefficients associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.

Figures

Figures reproduced from arXiv: 2606.07842 by the authors.

Figure 1
Figure 1. First row ((a)—(c)): heatmap of the BRW field B over the window [−R, R] 2 , for R = 100, 200, 400. Second row ((d)—(f)): heatmaps over the same windows of the Bargmann-Fock field F (with covariance E[F (0)F (x)] ≍ e −∥x∥ 2 ). One striking empirical feature of the two-dimensional BRW model is that numerical simu￾lations over large domains display characteristic quasi-linear filamentary patterns — typically described … view at source ↗
Figure 2
Figure 2. Several observables of BRW over the window [−R, R] 2 : Figures (a) and (b) correspond to R = 400 and are associated with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Nodal domains and nodal lines of BRW over the window [−R, R] 2 , for R = 200. They both correspond to the realization in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: On the left, a simulation of the mapping x 7→ h(φx), x ∈ [−100, 100]2 , where h ∼ FGF1/2(R 2 ), and where φx is an approximation of the Dirac mass at x, compactly supported in the ball B(x, ϱ), with ϱ = 0.07. On the right, a simulation of the mapping x 7→ W(φx), x ∈ [−…
Figure 5
Figure 5. Figure 5: One realization of a stationary Poisson line process with intensity t = 0.3, over the window [−R, R] 2 , for R = 100, 200, 400. Remark 2.3.3. It is important to notice that Poisson line processes are locally finite objects in the sense that, if ηt ∼ PLPt(R d ), then, w…
Figure 6
Figure 6. Figure 6: The singular random field from [11], as evoked in Remark 2.4.4-(i), over the window [−R, R] 2 , for R = 100, 700. (ii) We stress that the conclusions of Proposition 2.4.3 hold at a fixed level u, suggesting, in particular, that one mechanism contributing to the formati…
Figure 7
Figure 7. Figure 7: Simulation of the local maxima of BRW above the threshold u = 2 over the window [−R, R] 2 , for R = 1000, 2500 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Heatmap on the window [−R, R] 2 of a stationary Gaussian random field A = {A(x) : x ∈ R 2} with a smooth spectral measure supported on the annulus {x ∈ R 2 : 1.9 ≤ ∥x∥ ≤ 2.1}, for R = 100, 200, 400. (i) As R → ∞ and for fixed u, one has that, for some constants Cℓ , C′…

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Reference graph

Works this paper leans on

99 extracted references · 7 canonical work pages

  1. [1]

    Eigenfunctions and random waves in the Benjamini-Schramm limit.Preprint ArXiv: 1810.05601, October 2018

    Miklos Abert, Nicolas Bergeron, and Etienne Le Masson. Eigenfunctions and random waves in the Benjamini-Schramm limit.Preprint ArXiv: 1810.05601, October 2018. (Cited on p.2)

    work page arXiv 2018

  2. [2]

    R. J. Adler and J. E. Taylor.Random fields and geometry. Springer Monographs in Mathematics. Springer, New York, NY, 2007. (Cited on p.34, 35, 51)

    2007

  3. [3]

    Topological complexity of random sets

    Robert Adler, Jonathan Taylor, Shmuel Weinberger, and Steve Zelditch. Topological complexity of random sets. Technical report, American Institute of Mathematics, Palo Alto, CA, 2009. Workshop report, August 10–14, 2009. (Cited on p.3) SCARS IN RANDOM W A VES AND THEFGF( 1 2 )UNIVERSALITY CLASS 68

    2009

  4. [4]

    Notices of the american mathematical society, January 2008

    American Mathematical Society. Notices of the american mathematical society, January 2008. January 2008 issue. (Cited on p.3)

    2008

  5. [5]

    Zurich Lec- tures in Advanced Mathematics

    Nalini Anantharaman.Quantum ergodicity and delocalization of Schrödinger eigenfunctions. Zurich Lec- tures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, [2022]©2022. (Cited on p.2)

    2022

  6. [6]

    Zeros and critical points of Gaussian fields: Cumulants asymptotics and limit theorems.Preprint ArXiv: 2501.10226, 2025

    Michele Ancona, Louis Gass, Thomas Letendre, and Michele Stecconi. Zeros and critical points of Gaussian fields: Cumulants asymptotics and limit theorems.Preprint ArXiv: 2501.10226, 2025. (Cited on p.35)

    work page arXiv 2025

  7. [7]

    John Wiley & Sons, Inc., Hoboken, NJ, 2009

    Jean-Marc Azaïs and Mario Wschebor.Level sets and extrema of random processes and fields. John Wiley & Sons, Inc., Hoboken, NJ, 2009. (Cited on p.15, 16, 35)

    2009

  8. [8]

    Alex H. Barnett. Random plane waves, 2023.https://users.flatironinstitute.org/~ahb/rpws/(2009; updated 2023). (Cited on p.3)

    2023

Show all 99 references
  1. [9]

    Asymptotic rate of quantum ergodicity in chaotic euclidean billiards.Communications on Pure and Applied Mathematics, 59(10):1457–1488, 2006

    Alexander Barnett. Asymptotic rate of quantum ergodicity in chaotic euclidean billiards.Communications on Pure and Applied Mathematics, 59(10):1457–1488, 2006. (Cited on p.3, 5, 12)

    2006

  2. [10]

    Two point function for critical points of a random plane wave.Int

    Dmitry Beliaev, Valentina Cammarota, and Igor Wigman. Two point function for critical points of a random plane wave.Int. Math. Res. Not., 2019(9):2661–2689, 2019. (Cited on p.2)

    2019

  3. [11]

    Excursion fluctuations and spectral universality in gaussian fields,

    Dmitry Beliaev and Akshay Hegde. Excursion fluctuations and spectral universality in gaussian fields,

  4. [12]

    High local maxima of stationary smooth gaussian fields, 2026

    Dmitry Beliaev and Akshay Hegde. High local maxima of stationary smooth gaussian fields, 2026. (Cited on p.4, 15, 16)

    2026

  5. [13]

    Planck-scale distribution of nodal length of arithmetic random waves.J

    Jacques Benatar, Domenico Marinucci, and Igor Wigman. Planck-scale distribution of nodal length of arithmetic random waves.J. Anal. Math., 141(2):707–749, 2020. (Cited on p.2)

    2020

  6. [14]

    M. V. Berry. Regular and irregular semiclassical wavefunctions.J. Phys. A, 10(12):2083–2091, 1977. (Cited on p.2)

  7. [15]

    M. V. Berry. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature.J. Phys. A, 35(13):3025–3038, 2002. (Cited on p.2, 14, 20)

    2002

  8. [16]

    Generalized random fields and Lévy’s continuity the- orem on the space of tempered distributions.Commun

    Hermine Biermé, Olivier Durieu, and Yizao Wang. Generalized random fields and Lévy’s continuity the- orem on the space of tempered distributions.Commun. Stoch. Anal., 12(4):art. 4, 427–445, 2018. (Cited on p.7, 9, 23, 24, 63)

    2018

  9. [17]

    Self-similar random fields and rescaled random balls models.J

    Hermine Biermé, Anne Estrade, and Ingemar Kaj. Self-similar random fields and rescaled random balls models.J. Theor. Probab., 23(4):1110–1141, 2010. (Cited on p.8)

    2010

  10. [18]

    Blümel, I

    R. Blümel, I. H. Davidson, W. P. Reinhardt, H. Lin, and M. Sharnoff. Quasilinear ridge structures in water surface waves.Phys. Rev. A, 45:2641–2644, Feb 1992. (Cited on p.3, 6)

    1992

  11. [19]

    Bochner.Vorlesungen über Fouriersche Integrale

    S. Bochner.Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig, 1932. Reprinted by Chelsea, New York, 1948. (Cited on p.51)

    1932

  12. [20]

    Bojdecki, L

    T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. Limit theorems for occupation time fluctuations of branching systems. I: long-range dependence.Stochastic Processes Appl., 116(1):1–18, 2006. (Cited on p.8)

    2006

  13. [21]

    Bojdecki, L

    T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. Limit theorems for occupation time fluctuations of branchingsystems.II:Critical andlarge dimensions.Stochastic Processes Appl., 116(1):19–35, 2006. (Cited on p.8)

    2006

  14. [22]

    Brandolini, S

    L. Brandolini, S. Hofmann, and A. Iosevich. Sharp rate of average decay of the Fourier transform of a bounded set.Geom. Funct. Anal., 13(4):671–680, 2003. (Cited on p.31)

    2003

  15. [23]

    Monochromatic random waves for general Riemannian manifolds

    Yaiza Canzani. Monochromatic random waves for general Riemannian manifolds. InFrontiers in analy- sis and probability—in the spirit of the Strasbourg-Zürich meetings, pages 1–20. Springer, Cham, [2020] ©2020. (Cited on p.5, 18, 19)

    2020

  16. [24]

    Fixed frequency eigenfunction immersions and supremum norms of ran- dom waves.Electron

    Yaiza Canzani and Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of ran- dom waves.Electron. Res. Announc. Math. Sci., 22:76–86, 2015. (Cited on p.19)

    2015

  17. [25]

    Yaiza Canzani and Boris Hanin.C∞ scaling asymptotics for the spectral projector of the Laplacian.J. Geom. Anal., 28(1):111–122, 2018. (Cited on p.19)

    2018

  18. [26]

    Local universality for zeros and critical points of monochromatic random waves.Commun

    Yaiza Canzani and Boris Hanin. Local universality for zeros and critical points of monochromatic random waves.Commun. Math. Phys., 378(3):1677–1712, September2020.PublisherCopyright:©2020, Springer- Verlag GmbH Germany, part of Springer Nature. (Cited on p.2, 5, 18, 19)

    2020

  19. [27]

    Topology and nesting of the zero set components of monochromatic random waves.Comm

    Yaiza Canzani and Peter Sarnak. Topology and nesting of the zero set components of monochromatic random waves.Comm. Pure Appl. Math., 72(2):343–374, 2019. (Cited on p.19)

    2019

  20. [28]

    Fractional Gaussian forms and gauge theory: an overview.Front

    Sky Cao and Scott Sheffield. Fractional Gaussian forms and gauge theory: an overview.Front. Math. (Beijing), 21(1):1–137, 2026. (Cited on p.8, 25)

    2026

  21. [29]

    Leandro Chiarini, Milton Jara, and Wioletta M. Ruszel. Constructing fractional Gaussian fields from long-range divisible sandpiles on the torus.Stochastic Processes Appl., 140:147–182, 2021. (Cited on p.8) SCARS IN RANDOM W A VES AND THEFGF( 1 2 )UNIVERSALITY CLASS 69

    2021

  22. [30]

    YouriDavydovandVladimirRotar.Onasymptoticproximityofdistributions.J. Theor. Probab., 22(1):82– 98, erratum 99, 2009. (Cited on p.11)

    2009

  23. [31]

    Small scale CLTs for the nodal length of monochromatic waves.Comm

    Gauthier Dierickx, Ivan Nourdin, Giovanni Peccati, and Maurizia Rossi. Small scale CLTs for the nodal length of monochromatic waves.Comm. Math. Phys., 397(1):1–36, 2023. (Cited on p.2, 18, 19, 20, 21, 66)

    2023

  24. [32]

    Ap- proach to ergodicity in quantum wave functions.Phys

    Bruno Eckhardt, Shmuel Fishman, Jonathan Keating, Oded Agam, Jörg Main, and Kirsten Müller. Ap- proach to ergodicity in quantum wave functions.Phys. Rev. E, 52:5893–5903, Dec 1995. (Cited on p.12)

    1995

  25. [33]

    Kai-Tai Fang, Samuel Kotz, and Kai-Wang Ng.Symmetric multivariate and related distributions, vol- ume 36 ofMonogr. Stat. Appl. Probab.London etc.: Chapman and Hall, 1990. (Cited on p.33)

    1990

  26. [34]

    Federer.Geometric Measure Theory

    H. Federer.Geometric Measure Theory. Classics in Mathematics. Springer Berlin Heidelberg, 2014. (Cited on p.22, 34, 47)

    2014

  27. [35]

    Feichtinger and Thomas Strohmer, editors.Gabor Analysis and Algorithms: Theory and Appli- cations

    Hans G. Feichtinger and Thomas Strohmer, editors.Gabor Analysis and Algorithms: Theory and Appli- cations. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, 1998. (Cited on p.3)

    1998

  28. [36]

    Fernique

    X. Fernique. Généralisations du théorème de continuite de P. Levy.C. R. Acad. Sci., Paris, Sér. A, 266:25–28, 1968. (Cited on p.7, 9, 24)

    1968

  29. [37]

    A relation between two different formulations of the Berry’s conjecture.Preprint ArXiv: 2305.14906, 2023

    Alba García-Ruiz. A relation between two different formulations of the Berry’s conjecture.Preprint ArXiv: 2305.14906, 2023. (Cited on p.2)

    2023

  30. [38]

    Almost-sure asymptotics for Riemannian random waves.Bernoulli, 29(1):625 – 651, 2023

    Louis Gass. Almost-sure asymptotics for Riemannian random waves.Bernoulli, 29(1):625 – 651, 2023. (Cited on p.2)

    2023

  31. [39]

    Spectral criteria for the asymptotics of local functionals of Gaussian fields and their application to nodal volumes and critical points.Preprint ArXiv: 2501.07356, 2025

    Louis Gass. Spectral criteria for the asymptotics of local functionals of Gaussian fields and their application to nodal volumes and critical points.Preprint ArXiv: 2501.07356, 2025. (Cited on p.14, 36, 37, 38, 43)

    2025

  32. [40]

    Universal cancellations in uniform random waves, 2025

    Louis Gass, Domenico Marinucci, Giovanni Peccati, Francesca Pistolato, and Michele Stecconi. Universal cancellations in uniform random waves, 2025. (Cited on p.14, 19, 23)

    2025

  33. [41]

    The number of critical points of a Gaussian field: finiteness of moments

    Louis Gass and Michele Stecconi. The number of critical points of a Gaussian field: finiteness of moments. Probab. Theory Related Fields, 190(3-4):1167–1197, 2024. (Cited on p.35)

    2024

  34. [42]

    I. M. Gel’fand and N. Ya. Vilenkin.Generalized Functions. Vol. 4: Applications of Harmonic Analysis. Academic Press, New York, 1964. Translated from the Russian by Amiel Feinstein. (Cited on p.23)

    1964

  35. [43]

    P. R. Halmos and V. S. Sunder.Bounded integral operators onL 2 spaces, volume 96 ofErgeb. Math. Grenzgeb.Springer-Verlag, Berlin, 1978. (Cited on p.32)

    1978

  36. [44]

    Dphil thesis, University of Oxford, 2025

    Akshay Sateesh Hegde.Geometry of Smooth Gaussian Fields. Dphil thesis, University of Oxford, 2025. (Cited on p.4)

    2025

  37. [45]

    Birkhäuser Boston, Inc., Boston, MA, second edition, 1999

    Sigurdur Helgason.The Radon transform, volume 5 ofProgress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, second edition, 1999. (Cited on p.25, 48, 49, 51, 57, 59)

    1999

  38. [46]

    Eric J. Heller. Bound-state eigenfunctions of classically chaotic hamiltonian systems: Scars of periodic orbits.Phys. Rev. Lett., 53:1515–1518, Oct 1984. (Cited on p.6)

    1984

  39. [47]

    Eric J. Heller. Wavepacket dynamics and quantum chaology.Session LII-Chaos and quantum physics (Les Houches, France, 1989), 1991. (Cited on p.3, 6)

    1989

  40. [48]

    Heller.The semiclassical way to dynamics and spectroscopy

    Eric J. Heller.The semiclassical way to dynamics and spectroscopy. Princeton, NJ: Princeton University Press, 2018. (Cited on p.3, 6)

    2018

  41. [49]

    Springer-Verlag, Berlin, 2nd edition, 1990

    Lars Hörmander.The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, volume 256 ofGrundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 2nd edition, 1990. (Cited on p.50)

    1990

  42. [50]

    I: Distribution theory and Fourier analysis.Class

    Lars Hörmander.The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis.Class. Math. Berlin: Springer, reprint of the 2nd edition 1990 edition, 2003. (Cited on p.28)

    1990

  43. [51]

    DanielHugandRolfSchneider.Poisson hyperplane tessellations.SpringerMonogr.Math.Cham: Springer,

  44. [52]

    (Cited on p.10, 26, 27)

  45. [53]

    Local weak limits of Laplace eigenfunctions.Tunis

    Maxime Ingremeau. Local weak limits of Laplace eigenfunctions.Tunis. J. Math., 3(3):481–515, 2021. (Cited on p.2)

    2021

  46. [54]

    Tracts Math.Cambridge: Cambridge Uni- versity Press, 1997

    Svante Janson.Gaussian Hilbert spaces, volume 129 ofCamb. Tracts Math.Cambridge: Cambridge Uni- versity Press, 1997. (Cited on p.26, 35, 50, 51, 52, 53)

    1997

  47. [55]

    From beta-type distributions to high-dimensional volumes, volume 2383 ofLect

    Zakhar Kabluchko, David Albert Steigenberger, and Christoph Thäle.Random simplices. From beta-type distributions to high-dimensional volumes, volume 2383 ofLect. Notes Math.Cham: Springer, 2026. (Cited on p.8)

    2026

  48. [56]

    Kaplan and E.J

    L. Kaplan and E.J. Heller. Linear and nonlinear theory of eigenfunction scars.Annals of Physics, 264(2):171–206, 1998. (Cited on p.6)

    1998

  49. [57]

    arXiv preprint arXiv:1905.05136, 2019

    BlakeKeeler.Alogarithmicimprovementinthetwo-pointweyllawformanifoldswithoutconjugatepoints. arXiv preprint arXiv:1905.05136, 2019. (Cited on p.18, 20)

    1905

  50. [58]

    Nodal length fluctuations for arithmetic random waves.Ann

    Manjunath Krishnapur, Pär Kurlberg, and Igor Wigman. Nodal length fluctuations for arithmetic random waves.Ann. of Math. (2), 177(2):699–737, 2013. (Cited on p.2, 14) SCARS IN RANDOM W A VES AND THEFGF( 1 2 )UNIVERSALITY CLASS 70

    2013

  51. [59]

    (Cited on p.26, 27, 65)

    Günter Last and Mathew Penrose.Lectures on the Poisson process, volume 7 ofIMS Textb.Cambridge: Cambridge University Press, 2018. (Cited on p.26, 27, 65)

    2018

  52. [60]

    Asad Lodhia, Scott Sheffield, Xin Sun, and Samuel S. Watson. Fractional Gaussian fields: A survey. Probability Surveys, 13(none):1 – 56, 2016. (Cited on p.4, 8, 21, 24, 25, 26, 53)

    2016

  53. [61]

    Review of Yau’s conjecture on zero sets of Laplace eigen- functions

    Alexander Logunov and Eugenia Malinnikova. Review of Yau’s conjecture on zero sets of Laplace eigen- functions. InCurrent developments in mathematics 2018, pages 179–212. Int. Press, Somerville, MA, [2020] ©2020. (Cited on p.2)

    2018

  54. [62]

    Asymptotic covariances for functionals of weakly stationary random fields.Stochastic Processes Appl., 170:19, 2024

    Leonardo Maini. Asymptotic covariances for functionals of weakly stationary random fields.Stochastic Processes Appl., 170:19, 2024. Id/No 104297. (Cited on p.5, 12)

    2024

  55. [63]

    Spectral central limit theorem for additive functionals of isotropic and stationary gaussian fields.The Annals of Probability, 52(2):737–763, 2024

    Leonardo Maini and Ivan Nourdin. Spectral central limit theorem for additive functionals of isotropic and stationary gaussian fields.The Annals of Probability, 52(2):737–763, 2024. (Cited on p.12)

    2024

  56. [64]

    Non-universality of nodal length distribution for arithmetic random waves.Geom

    Domenico Marinucci, Giovanni Peccati, Maurizia Rossi, and Igor Wigman. Non-universality of nodal length distribution for arithmetic random waves.Geom. Funct. Anal., 26(3):926–960, 2016. (Cited on p.2, 14)

    2016

  57. [65]

    Domenico Marinucci, Maurizia Rossi, and Igor Wigman. The asymptotic equivalence of the sample trispec- trum and the nodal length for random spherical harmonics.Annales de l’Institut Henri Poincaré Proba- bilités et Statistiques, 56(1):374–390, 2020. (Cited on p.14)

    2020

  58. [66]

    R. E. Miles. Isotropic random simplices.Adv. Appl. Probab., 3:353–382, 1971. (Cited on p.8)

    1971

  59. [67]

    Nazarov and M

    F. Nazarov and M. Sodin. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions.Zh. Mat. Fiz. Anal. Geom., 12(3):205–278, 2016. (Cited on p.2)

    2016

  60. [68]

    On the number of nodal domains of random spherical harmonics.Am

    Fedor Nazarov and Mikhail Sodin. On the number of nodal domains of random spherical harmonics.Am. J. Math., 131(5):1337–1357, 2009. (Cited on p.2)

    2009

  61. [69]

    Integrating by spheres: Summary of blaschke-petkantschin formulas, 2019

    Anton Nikitenko. Integrating by spheres: Summary of blaschke-petkantschin formulas, 2019. (Cited on p.8)

    2019

  62. [70]

    StéphaneNonnenmacher.Anatomy of Quantum Chaotic Eigenstates, pages193–238.SpringerBasel, Basel,

  63. [71]

    Functional convergence of Berry’s nodal lengths: approximate tightness and total disorder.J

    Massimo Notarnicola, Giovanni Peccati, and Anna Vidotto. Functional convergence of Berry’s nodal lengths: approximate tightness and total disorder.J. Stat. Phys., 190(5):Paper No. 97, 41, 2023. (Cited on p.2, 14, 20, 66)

    2023

  64. [72]

    Nourdin, G

    I. Nourdin, G. Peccati, and M. Rossi. Nodal statistics of planar random waves.Comm. Math. Phys., 369(1):99–151, 2019. (Cited on p.2, 14, 20)

    2019

  65. [73]

    Tracts Math.Cambridge University Press, Cambridge, 2012

    Ivan Nourdin and Giovanni Peccati.Normal approximations with Malliavin calculus, volume 192 ofCamb. Tracts Math.Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. (Cited on p.5, 14, 26, 30, 31, 32, 35, 36, 51, 52, 67)

    2012

  66. [74]

    IvanNourdinandGiovanniPeccati.Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Camb. Tracts Math. Cambridge University Press, 2012. (Cited on p.51, 52)

    2012

  67. [75]

    Nodal statistics of planar random waves.Comm

    Ivan Nourdin, Giovanni Peccati, and Maurizia Rossi. Nodal statistics of planar random waves.Comm. Math. Phys., 369(1):99–151, 2019. (Cited on p.14)

    2019

  68. [76]

    Multivariate normal approximation on the Wiener space: new bounds in the convex distance.J

    Ivan Nourdin, Giovanni Peccati, and Xiaochuan Yang. Multivariate normal approximation on the Wiener space: new bounds in the convex distance.J. Theor. Probab., 35(3):2020–2037, 2022. (Cited on p.7)

    2020

  69. [77]

    Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2009

    David Nualart.Malliavin calculus and its applications, volume 110 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2009. (Cited on p.35)

    2009

  70. [78]

    O’Connor, J

    P. O’Connor, J. Gehlen, and E. J. Heller. Properties of random superpositions of plane waves.Phys. Rev. Lett., 58:1296–1299, Mar 1987. (Cited on p.3, 4, 6)

    1987

  71. [79]

    The Leray measure of nodal sets for random eigen- functions on the torus.Journal de l’Institut Fourier, 1(58):299–335, 2008

    Ferenc Oravzcek, Zeév Rudnick, and Igor Wigman. The Leray measure of nodal sets for random eigen- functions on the torus.Journal de l’Institut Fourier, 1(58):299–335, 2008. (Cited on p.2)

    2008

  72. [80]

    Peccati, J

    G. Peccati, J. L. Solé, M. S. Taqqu, and F. Utzet. Stein’s method and normal approximation of Poisson functionals.Ann. Probab., 38(2):443–478, 2010. (Cited on p.66)

    2010

  73. [81]

    Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry, volume 7 ofBocconi Springer Ser.Milano: Bocconi University Press; Cham: Springer, 2016

    Giovanni Peccati and Matthias Reitzner, editors.Stochastic analysis for Poisson point processes. Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry, volume 7 ofBocconi Springer Ser.Milano: Bocconi University Press; Cham: Springer, 2016. (Cited on p.26)

    2016

  74. [82]

    Taqqu.Wiener chaos: moments, cumulants and diagrams, volume 1 ofBocconi & Springer Series

    Giovanni Peccati and Murad S. Taqqu.Wiener chaos: moments, cumulants and diagrams, volume 1 ofBocconi & Springer Series. Springer, Milan; Bocconi University Press, Milan, 2011. A survey with computer implementation, Supplementary material available online. (Cited on p.7)

    2011

  75. [83]

    Gaussian random measures generated by Berry’s nodal sets.J

    Giovanni Peccati and Anna Vidotto. Gaussian random measures generated by Berry’s nodal sets.J. Stat. Phys., 178(4):996–1027, 2020. (Cited on p.2, 14, 19, 20) SCARS IN RANDOM W A VES AND THEFGF( 1 2 )UNIVERSALITY CLASS 71

    2020

  76. [84]

    Local Laws and Fluctuations for Super-Coulombic Riesz Gases

    Luke Peilen and Sylvia Serfaty. Local Laws and Fluctuations for Super-Coulombic Riesz Gases. Preprint, arXiv:2511.18623 [math-ph] (2025), 2025. (Cited on p.8)

    2025

  77. [85]

    Ranga Rao

    R. Ranga Rao. Relations between Weak and Uniform Convergence of Measures with Applications.The Annals of Mathematical Statistics, 33(2):659 – 680, 1962. (Cited on p.66)

    1962

  78. [86]

    The behaviour of eigenstates of arithmetic hyperbolic manifolds.Com- mun

    Zeév Rudnick and Peter Sarnak. The behaviour of eigenstates of arithmetic hyperbolic manifolds.Com- mun. Math. Phys., 161(1):195–213, 1994. (Cited on p.6)

    1994

  79. [87]

    On the volume of nodal sets for eigenfunctions of the Laplacian on the torus.Ann

    Zeév Rudnick and Igor Wigman. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus.Ann. Henri Poincaré, 9(1):109–130, 2008. (Cited on p.2)

    2008

  80. [88]

    Sarnak and I

    P. Sarnak and I. Wigman. Topologies of nodal sets of random band-limited functions.Comm. Pure Appl. Math., 72(2):275–342, 2019. (Cited on p.19)

    2019

  81. [89]

    Rolf Schneider and Wolfgang Weil.Stochastic and integral geometry. Probab. Appl. Berlin: Springer, 2008. (Cited on p.10)

    2008

  82. [90]

    Hermann, Paris, new edition, 1966

    Laurent Schwartz.Théorie des distributions. Hermann, Paris, new edition, 1966. (Cited on p.50)

    1966

  83. [91]

    Gaussian free fields for mathematicians.Probab

    Scott Sheffield. Gaussian free fields for mathematicians.Probab. Theory Relat. Fields, 139(3-4):521–541,

  84. [92]

    Sogge and Steve Zelditch

    Christopher D. Sogge and Steve Zelditch. Riemannian manifolds with maximal eigenfunction growth.Duke Math. J., 114(3):387–437, 2002. (Cited on p.19)

    2002

  85. [93]

    New chaos decomposition of Gaussian nodal volumes.Preprint ArXiv: 2505.22350, 2025

    Michele Stecconi and Anna Paola Todino. New chaos decomposition of Gaussian nodal volumes.Preprint ArXiv: 2505.22350, 2025. (Cited on p.2, 14)

    2025

  86. [94]

    Filament structure of random waves.Nonlinearity, 36(3):1547–1570, 2023

    Melissa Tacy. Filament structure of random waves.Nonlinearity, 36(3):1547–1570, 2023. (Cited on p.3)

    2023

  87. [95]

    Holder continuity of interfaces for scale-invariant poisson stick soup, 2026

    Augusto Teixeira and Daniel Ungaretti. Holder continuity of interfaces for scale-invariant poisson stick soup, 2026. (Cited on p.11)

    2026

  88. [96]

    I. Wigman. Fluctuations of the nodal length of random spherical harmonics.Commun. Math. Phys., 298(3):787–831, Jun 2010. (Cited on p.2, 14)

    2010

  89. [97]

    On the nodal structures of random fields: a decade of results.J

    Igor Wigman. On the nodal structures of random fields: a decade of results.J. Appl. Comput. Topol., 8(6):1917–1959, 2024. (Cited on p.2)

    1917

  90. [98]

    102 ofAnnals of Mathemat- ics Studies

    Shing Tung Yau, editor.Seminar on Differential Geometry, volume No. 102 ofAnnals of Mathemat- ics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. (Cited on p.2)

    1982

  91. [99]

    Real and complex zeros of Riemannian random waves

    Steven Morris Zelditch. Real and complex zeros of Riemannian random waves. In Motoko Kotani, Hisashi Naito, and Tatsuya Tate, editors,Spectral Analysis in Geometry and Number Theory, Contemp. Math., pages 321–342. American Mathematical Society, 2009. (Cited on p.5, 18, 19)

    2009

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.