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Statistics on Yau's conjecture: Variance asymptotics

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Pith's one-line read On manifolds without conjugate points, monochromatic random waves have nodal-volume variance that decays like ℓ^{1-n}/log ℓ, more than squaring the previous bound and forcing Berry cancellation.

desk verdict Solid monochromatic variance improvement on no-conjugate-point manifolds, with reusable abstract machinery; residual gap to sharp rate is openly left open. read the letter →

arxiv 2607.09946 v1 pith:YOAGRD4W submitted 2026-07-10 math.PR math-phmath.DGmath.MP

Michele Stecconi , Anna Paola Todino This is my paper
classification math.PRmath-phmath.DGmath.MP MSC 60G6058J5035P20
keywords YauconjecturenodalvolumeRiemannianrandomwavesmonochromaticregimeBerrycancellationpointwiseWeyllawWienerchaosKac-Riceformula
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

Yau’s conjecture predicts that the nodal volume of a Laplace eigenfunction of frequency λ is of order λ. This paper studies the probabilistic version: the normalized nodal volume Y_M(I) of a Riemannian random wave built from eigenfunctions whose frequencies lie in a spectral window I of height ℓ. The authors prove that for any non-monochromatic window the variance of Y_M is exactly of order ℓ^{-n}, while on manifolds without conjugate points the pure monochromatic window of width 1 yields the sharper upper bound O(ℓ^{1-n}/log ℓ). That rate is more than the square of the earlier Canzani–Hanin bound and is already smaller than 1 over the dimension of the eigenspace, which is precisely Berry’s cancellation. The proof rests on a new abstract variance machine (Theorem C) that converts near-diagonal Kac–Rice estimates and off-diagonal chaos estimates into a global Law of Large Numbers once a scaling limit of the covariance holds in a fixed macroscopic range; the required scaling limit is supplied by Keeler’s logarithmic improvement of the pointwise Weyl law.

What carries the argument

Theorem C: an abstract quantitative Law of Large Numbers for the nodal volume of an arbitrary C^3 Gaussian field, controlled solely by a near-diagonal C^3 bound and an off-diagonal short-correlation bound on the covariance jet; once a macroscopic scaling limit of the random-wave covariance is available, the theorem yields the stated variance rates.

What would settle it

Compute or rigorously bound the C^1 scaling error δ_M of the monochromatic spectral projector on a manifold without conjugate points in a fixed positive range; if the error remains only O(1) rather than O(1/log ℓ), the monochromatic variance upper bound collapses to the weaker Canzani–Hanin order.

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Extended reading notes

Core claim

On every compact manifold without conjugate points the monochromatic random wave ϕ_{[ℓ−1,ℓ]} satisfies Var(Y_M([ℓ−1,ℓ])) = O(ℓ^{1-n}/log ℓ). Consequently the variance is o(1/N_M), i.e., Berry cancellation occurs, while for every non-monochromatic window the variance is exactly Θ(ℓ^{-n}).

Load-bearing premise

The monochromatic covariance must admit a scaling limit with error O(1/log ℓ) on balls of fixed positive radius, which is supplied only by Keeler’s logarithmic refinement of the pointwise Weyl law and fails as soon as that logarithmic gain is lost.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the high-frequency variance of the normalized nodal volume Y_M(I_ℓ) of Riemannian random waves on a compact manifold, for spectral windows I_ℓ of various widths (full, weakly monochromatic, and monochromatic). Theorem A gives exact order Θ(ℓ^{-n}) for non-monochromatic windows (α<1) with no geometric assumptions, and the improved upper bound O(ℓ^{1-n}/log ℓ) for monochromatic windows on manifolds without conjugate points (Corollary 3.4.3). This improves the Canzani–Hanin O(ℓ^{-(n-1)/2}) bound by more than a square and implies Berry cancellation (Var = o(1/N)). The proofs combine Kac–Rice near the diagonal, the authors’ Wiener–Itô chaos expansion from the companion paper [44], and a new analysis of the scaling error in the pointwise Weyl law for arbitrary windows (Theorem B), all fed into an abstract variance machine (Theorem C) that requires only near- and off-diagonal covariance bounds.

Significance. The monochromatic variance bound on manifolds without conjugate points is a genuine advance: it is the first result that reaches spectral windows of size O(1) on general chaotic geometries and yields a form of Berry cancellation that was previously known only for spheres, tori, and Euclidean space. The abstract machinery of Theorem C and the scaling-error reduction of Theorem B are reusable tools that cleanly separate geometry (Keeler’s logarithmic Weyl-law improvement) from probabilistic estimates. The residual factor ℓ/log ℓ to the conjectured Θ(ℓ^{-n}) rate is openly acknowledged, so the claimed upper bound is not overstated. The logical chain is fully written and the only external analytic input (Keeler) is correctly cited.

minor comments (4)
  1. The notation for the scaling error δ_M_I(α,C^k,[ρ',ρ]) is introduced in Definition 3.5.2 and then immediately abbreviated; a short reminder of the range and order of derivatives when the abbreviation is first used in Theorem B would help the reader.
  2. In the statement of Theorem C the constant T_0 is described only as “big enough (e.g. 77)”; a brief indication of how it arises from the eccentricity and short-correlation thresholds would make the quantitative claim more transparent.
  3. A few typographical slips remain (e.g. “Acknowlegdments”, “Hoermander”, “varaince”). They do not affect readability but should be corrected.
  4. The comparison with the sphere (Section 3.8) is clear, yet a one-sentence pointer to the precise rate known for spherical harmonics (Eq. (3.2)) already in the introduction would orient the reader earlier.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation of companion chaos expansion; no reduction of claims to inputs by construction.

  1. self citation load bearing [Abstract; §3.3; proof of Theorem C (esp. Step VIII / Thm 5.2.1); §7.4–7.5]
    "Our proofs rely on a local and global analysis combining the Kac-Rice formula, the new Wiener-Itô chaos decompositions of [44] … Theorem 5.2.1 (Theorem 1.10 [44]). Let f:M o R be as above. Then |E{L_f(dx)[q]·L_f(dy)[q]}|≤2^q·λ(f,x)λ(f,y)/n·∥j^{1,1}_{x,y}C∥^q_{g_f} dx dy."

    The off-diagonal control that converts short-range correlation assumptions into a variance upper bound is taken verbatim from the authors’ companion paper [44]. This is load-bearing for the method (without it the fourth-chaos integral cannot be estimated), yet the cited identity is a general, parameter-free expansion that does not presuppose the random-wave variance rates; the geometric input (Keeler scaling error) remains external and independent. Hence only minor self-citation, not a circular reduction of the claim to its own input.

full rationale

The derivation chain is self-contained and non-circular. Theorem C is an abstract quantitative LLN for nodal volume of an arbitrary C^3 Gaussian field, proved from Kac-Rice (near-diagonal) plus the Wiener-Itô expansion of the companion [44] (off-diagonal fourth-and-higher chaos). Theorem B reduces the scaling error δ_M of an arbitrary spectral window to the Euclidean error plus the classical pointwise Weyl remainder of [0,ℓ]; the monochromatic case then plugs in Keeler’s external logarithmic improvement (no conjugate points). Theorem A and the Berry-cancellation corollaries follow by feeding those δ bounds into the abstract machine of Theorem C / Prop. 3.6.1. The only self-citation is [44], which supplies a general analytic identity (chaos expansion valid for any Gaussian field on any manifold) whose hypotheses do not include the variance rates claimed here; it is therefore independent support, not a circular reduction. No fitted parameters, no uniqueness theorems imported from the authors, no ansatz smuggled via citation, and no renaming of known empirical patterns. Residual gap to the conjectured Θ(ℓ^{-n}) monochromatic rate is openly acknowledged and does not affect the claimed upper bounds.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper is asymptotic pure mathematics. It inherits standard Riemannian geometry, Gaussian analysis and microlocal analysis; the only non-classical analytic input is Keeler’s logarithmic Weyl-law improvement and the authors’ own chaos expansion. No free parameters are fitted.

assumptions (5)
  • domain assumption Keeler’s logarithmic improvement of the pointwise Weyl law on manifolds without conjugate points: δ_M_{[0,ℓ]} = O(1/(ℓ log ℓ)) in fixed range (Thm 3.5.3(3)).
    Load-bearing for the monochromatic scaling limit (Thm B and Cor 3.5.5); without it the monochromatic variance bound reverts to the weaker Canzani–Hanin order.
  • domain assumption Wiener–Itô chaos expansion of the nodal volume for arbitrary C^3 Gaussian fields on Riemannian manifolds (Stecconi–Todino 2025).
    Used throughout to control the off-diagonal contribution (Thm C, Step VIII); taken as a black box from the companion paper.
  • standard math Hörmander’s classical pointwise Weyl law with O(ℓ^{-1}) remainder on any compact manifold.
    Baseline for the colourful and weakly monochromatic regimes (Thm 2.5.6).
  • standard math Standard Kac–Rice formula for the second moment of the nodal measure of a non-degenerate C^3 Gaussian field.
    Used for the near-diagonal contribution (Lemma 5.1.3).
  • standard math Bessel-function decay estimates |∇^α K(u)| ≲ (1+|u|)^{-(n-1)/2} for the Euclidean covariance kernels.
    Appendix A; needed for the Euclidean scaling error (Thm 7.2.2) and for the fourth-chaos integral.
invented entities (2)
  • C^k scaling error δ_M_I(α,C^k,[ρ',ρ]) of a random wave (Def 3.5.2) independent evidence
    purpose: Quantifies the difference between the manifold covariance and its Euclidean counterpart uniformly in derivatives and in a prescribed range; the vanishing of this quantity is the hypothesis that feeds Theorem C.
    A convenient bookkeeping device rather than a new physical object; its asymptotic behaviour is derived from classical Weyl laws.
  • Abstract variance machine (Theorem C) independent evidence
    purpose: Isolates two local covariance conditions (near-diagonal C^3 bound + off-diagonal short-range C^1 bound) that force the nodal-volume variance to decay.
    A reusable analytic criterion; its correctness is proved in the paper and does not rely on random-wave structure.

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Pith. "Pith review of Statistics on Yau's conjecture: Variance asymptotics." pith.science (2026). https://pith.science/paper/YOAGRD4W

@misc{pith2026260709946,
  author       = {Pith},
  title        = {Pith review of: Statistics on Yau's conjecture: Variance asymptotics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOAGRD4W}},
  note         = {Machine review of arXiv:2607.09946}
}
read the original abstract

We investigate the probabilistic counterpart of Yau's conjecture on the nodal volume of Laplace eigenfunctions on compact manifolds, by studying the high-frequency variance asymptotics of Riemannian random waves. We establish (Theorem A) a quantitative bound for the fluctuations of their nodal volumes, depending on different regimes of spectral windows, including the monochromatic one: with spectral size 1. Notably, our bounds improve, by more than a power 2, the existing results in the literature, cf. Canzani and Hanin (2020), in the case of manifolds without conjugate pairs, in particular negatively curved ones. As a corollary, we prove that Berry's cancellation phenomenon occurs for monochromatic Riemannian Random Waves on such chaotic manifolds. Our proofs rely on a local and global analysis combining the Kac-Rice formula, the new Wiener-It\^o chaos decompositions of Stecconi and Todino (2025), and a sharp analysis of the error in the pointwise Weyl law associated to arbitrary spectral window (Theorem B). We introduce a general machinery (Theorem C), which ensures variance decay under broad geometric conditions, subject to correlation decay assumptions.

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Works this paper leans on

56 extracted references · 2 linked inside Pith

  1. [44]

    Stecconi and A

    M. Stecconi and A. P. Todino. New chaos decomposition of gaussian nodal volumes.arXiv:2505.22350,

    arXiv

  2. [1]

    Abramowitz.Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,

    M. Abramowitz.Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,. Dover Publications, Inc., USA, 1974. (Cited on p.44)

    1974

  3. [2]

    R. J. Adler and J. E. Taylor.Random fields and geometry. Springer Monographs in Mathematics. Springer, New York, 2007. (Cited on p.3, 19, 21)

    2007

  4. [3]

    Azais and M

    J.-M. Azais and M. Wschebor.Level sets and extrema of random processes and fields. John Wiley & Sons, Inc., Hoboken, NJ, 2009. (Cited on p.3, 21)

    2009

  5. [4]

    Beliaev and I

    D. Beliaev and I. Wigman. Volume distribution of nodal domains of random band-limited functions. Probability Theory and Related Fields, 172(1):453–492, 2018. (Cited on p.8, 9)

    2018

  6. [5]

    Benatar, D

    J. Benatar, D. Marinucci, and I. Wigman. Planck-scale distribution of nodal length of arithmetic random waves.Journal d’Analyse Mathématique, 141(2):707–749, 2020. (Cited on p.21)

    2020

  7. [6]

    P. Berard. Volume des ensembles nodaux des fonctions propres du Laplacien.BonySjostrand-Meyer sem- inar, 1984-1985, Exp. No 14, 10 pp., Ecole Polytech. Palaiseau., 1985. (Cited on p.9)

    1984

  8. [7]

    M. V. Berry. Regular and irregular semiclassical wavefunctions.Journal of Physics. A. Mathematical and General, 10(12):2083–2091, 1977. (Cited on p.8, 9) STATISTICS ON YAU’S CONJECTURE 46

Show all 56 references
  1. [8]

    M. V. Berry. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature.Journal of Physics. A. Mathematical and General, 35(13):3025–3038, 2002. (Cited on p.9)

    2002

  2. [9]

    Billingsley.Convergence of probability measures

    P. Billingsley.Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication. (Cited on p.2)

    1999

  3. [10]

    Bogachev.Gaussian Measures

    V. Bogachev.Gaussian Measures. Mathematical surveys and monographs. American Mathematical Soci- ety, 1998. (Cited on p.3)

    1998

  4. [11]

    Cammarota

    V. Cammarota. Nodal area distribution for arithmetic random waves.Transactions of the American Math- ematical Society, 2017. (Cited on p.9)

    2017

  5. [12]

    Y. Canzani. Monochromatic random waves for general Riemannian manifolds. InFrontiers in analysis and probability—in the spirit of the Strasbourg-Zürich meetings, pages 1–20. Springer, Cham, [2020]©2020. (Cited on p.7, 8, 13)

    2020

  6. [13]

    Canzani and B

    Y. Canzani and B. Hanin.C∞ scaling asymptotics for the spectral projector of the Laplacian.Journal of Geometric Analysis, 28(1):111–122, 2018. (Cited on p.6, 8, 14, 15)

    2018

  7. [14]

    Canzani and B

    Y. Canzani and B. Hanin. Local universality for zeros and critical points of monochromatic random waves. Communications In Mathematical Physics, 378(3):1677–1712, Sept. 2020. Publisher Copyright:©2020, Springer-Verlag GmbH Germany, part of Springer Nature. (Cited on p.1, 6, 7,...

    2020

  8. [15]

    Canzani and P

    Y. Canzani and P. Sarnak. Topology and nesting of the zero set components of monochromatic random waves.Communications on Pure and Applied Mathematics, 72(2):343–374, 2019. (Cited on p.7)

    2019

  9. [16]

    Dalmao, A

    F. Dalmao, A. Estrade, and J. León. On 3-dimensional Berry’s model.Latin American Journal of Proba- bility and Mathematical Statistics, 18:377, 01 2021. (Cited on p.9)

    2021

  10. [17]

    Dierickx, I

    G. Dierickx, I. Nourdin, G. Peccati, and M. Rossi. Small scale clts for the nodal length of monochromatic waves.Communications in Mathematical Physics, 397:1–36, 2023. (Cited on p.8, 9, 10, 17)

    2023

  11. [18]

    Donnelly

    H. Donnelly. Bounds for eigenfunctions of the laplacian on compact riemannian manifolds.Journal of Functional Analysis, 187(1):247–261, 2001. (Cited on p.10)

    2001

  12. [19]

    Donnelly and C

    H. Donnelly and C. Fefferman. Nodal sets of eigenfunctions on Riemannian manifolds.Inventiones Math- ematicae, 93(1):161–183, 1988. (Cited on p.4)

    1988

  13. [20]

    L. Gass. Almost-sure asymptotics for Riemannian random waves.Bernoulli, 29(1):625 – 651, 2023. (Cited on p.6, 8, 9)

    2023

  14. [21]

    L. Gass. Spectral criteria for the asymptotics of local functionals of Gaussian fields and their application to nodal volume, 2025. (Cited on p.9)

    2025

  15. [22]

    Gass and M

    L. Gass and M. Stecconi. The number of critical points of a Gaussian field: finiteness of moments.Proba- bility Theory and Related Fields, Apr 2024. (Cited on p.21)

    2024

  16. [23]

    Grotto, L

    F. Grotto, L. Maini, and A. P. Todino. Fluctuations of polyspectra in spherical and Euclidean random wave models.Electronic Communications in Probability, 29:Paper No. 9, 12, 2024. (Cited on p.29, 40)

    2024

  17. [24]

    Hörmander

    L. Hörmander. The spectral function of an elliptic operator.Acta Mathematica, 121(none):193 – 218, 1968. (Cited on p.6, 13)

    1968

  18. [25]

    Ingremeau

    M. Ingremeau. Volume d’ensembles nodaux de fonctions propres du Laplacien [d’après Logunov, Malin- nikova, ainsi que Yau, Brüning, Donnelly-Feffermann, Hardt-Simon, ...]. Number 430, pages Exp. No. 1172, 183–214. 2021. Séminaire Bourbaki. Vol. 2019/2021. Exposés 1166–1180. (Ci...

    2021

  19. [26]

    B. Keeler. A logarithmic improvement in the two-point Weyl Law for manifolds without conjugate points. Annales de l’Institut Fourier, 74(2):719–762, 2024. (Cited on p.5, 6, 7, 8, 9, 13, 14, 15)

    2024

  20. [27]

    Krasikov

    I. Krasikov. Approximations for the bessel and airy functions with an explicit error term.LMS Journal of Computation and Mathematics, 17(1):209–225, 2014. (Cited on p.40)

    2014

  21. [28]

    J. M. Lee.Introduction to Riemannian manifolds, volume 176 ofGraduate Texts in Mathematics. Springer, Cham, second edition, 2018. (Cited on p.18)

    2018

  22. [29]

    Lerario and M

    A. Lerario and M. Stecconi. Differential topology of Gaussian random fields.arXiv:1902.03805, 2019. (Cited on p.21)

    1902 arXiv

  23. [30]

    A. Logunov. Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure. Annals of Mathematics (2), 187(1):221–239, 2018. (Cited on p.4)

    2018

  24. [31]

    A. Logunov. Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture.Annals of Mathematics (2), 187(1):241–262, 2018. (Cited on p.4)

    2018

  25. [32]

    Logunov and E

    A. Logunov and E. Malinnikova. Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three. In50 years with Hardy spaces, volume 261 ofOper. Theory Adv. Appl., pages 333–344. Birkhäuser/Springer, Cham, 2018. (Cited on p.4)

    2018

  26. [33]

    Logunov and E

    A. Logunov and E. Malinnikova. Review of Yau’s conjecture on zero sets of Laplace eigenfunctions. In Current developments in mathematics 2018, pages 179–212. Int. Press, Somerville, MA, [2020]©2020. (Cited on p.4) STATISTICS ON YAU’S CONJECTURE 47

    2018

  27. [34]

    Marinucci, G

    D. Marinucci, G. Peccati, M. Rossi, and I. Wigman. Non-universality of nodal length distribution for arithmetic random waves.Geometric and Functional Analysis, 26(3):926–960, 2016. (Cited on p.9)

    2016

  28. [35]

    (Cited on p.9, 17)

    D.Marinucci, M.Rossi, andA.P.Todino.Laguerreexpansionfornodal volumesandapplications.INdAM- Springer Volume, Analysis and Geometry of Random Field, 2026. (Cited on p.9, 17)

    2026

  29. [36]

    Marinucci, M

    D. Marinucci, M. Rossi, and I. Wigman. The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics.Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 56(1):374–390, 2020. (Cited on p.9, 17, 21)

    2020

  30. [37]

    Marinucci and M

    D. Marinucci and M. Stecconi. Critical Points of Chi-Fields.ALEA. Latin American Journal of Probability and Mathematical Statistics, 22(1):749–773, 2025. (Cited on p.31)

    2025

  31. [38]

    Mathis and M

    L. Mathis and M. Stecconi. Expectation of a random submanifold: the zonoid section.Annales Henri Lebesgue, 7:903–967, 2024. (Cited on p.31)

    2024

  32. [39]

    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.8, 21)

    2016

  33. [40]

    Nourdin, G

    I. Nourdin, G. Peccati, and M. Rossi. Nodal statistics of planar random waves.Communications in Math- ematical Physics, 369(1):99–151, 2019. (Cited on p.9)

    2019

  34. [41]

    Peccati and M

    G. Peccati and M. Stecconi. Nodal volumes as differentiable functionals of Gaussian fields.Transactions of the American Mathematical Society, Published electronically, May 2025. (Cited on p.21)

    2025

  35. [42]

    Sarnak and I

    P. Sarnak and I. Wigman. Topologies of nodal sets of random band-limited functions.Communications on Pure and Applied Mathematics, 72(2):275–342, 2019. (Cited on p.7, 8, 15)

    2019

  36. [43]

    C. D. Sogge and S. Zelditch. Focal points and sup-norms of eigenfunctions.Revista Matemática Iberoamer- icana,, 32(3):971–994, 2016. (Cited on p.10)

    2016

  37. [45]

    (Cited on p.1, 3, 5, 6, 7, 9, 11, 12, 16, 20, 23, 24, 25, 41, 42, 44, 45)

  38. [46]

    A. P. Todino. No smooth phase transition for the nodal length of band-limited spherical random fields. Stochastic Processes and their Applications, 169:104273, 2024. (Cited on p.9, 17)

    2024

  39. [47]

    A. Vidotto. A note on the reduction principle for the nodal length of planar random waves.Statistics & Probability Letters, 174:109090, 2021. (Cited on p.9)

    2021

  40. [48]

    H. Weyl. Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen (mit einer anwendung auf die theorie der hohlraumstrahlung).Mathematische Annalen, 71(4):441–479, 1912. (Cited on p.5)

    1912

  41. [49]

    I. Wigman. On the distribution of the nodal sets of random spherical harmonics.Journal of mathematical physics, 50(1):013521, 2009. (Cited on p.9, 17)

    2009

  42. [50]

    I. Wigman. Fluctuations of the nodal length of random spherical harmonics.Communications in Mathe- matical Physics, 298(3):787, 2010. (Cited on p.9, 21)

    2010

  43. [51]

    I. Wigman. On the expected Betti numbers of the nodal set of random fields.Analysis & pde, 3 2020. (Cited on p.7, 8, 15)

    2020

  44. [52]

    I. Wigman. On the nodal structures of random fields: a decade of results.Journal of Applied and Com- putational Topology, pages 1–43, 2023. (Cited on p.7, 8, 9, 10, 15, 38)

    2023

  45. [53]

    S. T. Yau, editor.Seminar on Differential Geometry, volume No. 102 ofAnnals of Mathematics 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.4)

    1982

  46. [54]

    S.-T. Yau. Open problems in geometry. InDifferential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), volume 54, Part 1 ofProc. Sympos. Pure Math., pages 1–28. Amer. Math. Soc., Providence, RI, 1993. (Cited on p.4)

    1990

  47. [55]

    Zelditch

    S. Zelditch. Fine structure of zoll spectra.Journal of Functional Analysis, 143(2):415–460, 1997. (Cited on p.17)

    1997

  48. [56]

    Zelditch

    S. Zelditch. Real and complex zeros of Riemannian random waves. In M. Kotani, H. Naito, and T. Tate, editors,Spectral Analysis in Geometry and Number Theory, Contemporary Mathematics, pages 321–342. American Mathematical Society, 2009. (Cited on p.5, 7, 8, 9)

    2009

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