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Hyperuniform random measures, transport and rigidity

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Pith's one-line read This survey establishes that hyperuniformity of a stationary random measure—variance in large balls growing slower than volume—is equivalent to the vanishing of its spectral measure at the origin, and that this one spectral condition contro

desk verdict A well-executed survey, not a new-results paper, whose central spectral theorem leans on an unproved lemma quoted from the author's own preprint; needs to either prove it or clearly flag pending results before it can serve as a reference. read the letter →

arxiv 2510.18392 v2 pith:MKRNB522 submitted 2025-10-21 math.PR

Raphaël Lachièze-Rey (MAP5 - UMR 8145 , MATHNET) This is my paper
classification math.PR MSC 60G5760G5560D0560B20
keywords hyperuniformityrandommeasuresspectralmeasurestructurefactorrigidityoptimaltransportdeterminantalpointprocessesperturbedlattices
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 survey tries to establish that hyperuniformity—a stationary random measure whose mass in large balls fluctuates sublinearly in volume—is not a special property of lattices but a second-order spectral condition: the structure factor must vanish at the origin fast enough. Its central theorem makes this precise: number variance on balls, subextensive variance of any smooth linear statistic with nonzero integral, and S(B_ε)/ε^d → 0 are equivalent for wide-sense stationary random measures. The same spectral quantity then drives macroscopic consequences: hyperuniform processes in dimension 2 admit optimal-transport matchings to lattices; higher hyperuniformity exponents produce rigidity of moments; stealthy processes (spectrum zero near the origin) are maximally rigid. A sympathetic reader would care because it turns a scattering-measurement observable into a rigorous organizing principle for particle systems, random matrices, zeros of random functions, and quasicrystals.

What carries the argument

The spectral measure S, which encodes how variance distributes over frequencies via the Plancherel identity, is the workhorse. The paper repeatedly converts variance estimates into statements about S near zero and at infinity, using the scaling f_R(x)=f(x/R), the explicit oscillatory Fourier transform of the unit-ball indicator, and the growth bound that restricts how fast S can grow at large frequencies. The rigidity criterion in Theorem 5.1 is the same spectral quantity: whether the integral of |u|^{2k}/s(u) converges or diverges separates k-rigid from non-rigid models.

What would settle it

Take a stationary Gaussian random measure with spectral measure a weighted sum of Dirac masses accumulating at the origin (for example Σ k^{-2} δ_{1/k}), normalize it, and compute both S(B_ε)/ε^d and Var(M(B_R)) using the explicit Fourier transform of the unit-ball indicator. If S(B_ε)/ε^d → 0 but Var(M(B_R)) is not o(R^d), Theorem 2.1's spectral-to-variance direction fails; conversely, if the growth bound on S fails, Lemma 2.2 is false.

Watch

Extended reading notes

Core claim

The load-bearing assertion is Theorem 2.1: for a wide-sense stationary random measure with spectral measure S, hyperuniformity (Var M(B_R)=o(R^d)) is equivalent to Var M(f_R)=o(R^d) for some admissible test function f with nonzero integral, and to spectral hyperuniformity S(B_ε)=o(ε^d). The proof hinges on the Fourier representation of covariance and on a spectral growth bound for S. The survey then derives quantitative refinements—the hyperuniformity exponent α determines how fast smooth-statistic variances decay (Proposition 2.1), a universal lower bound forces number variance to be at least R^{d-1}, and rigidity of order k is characterized by divergence of the integral of |u|^{2k}/s(u) wh

Load-bearing premise

The equivalence theorem rests on a spectral growth bound (Lemma 2.2) whose proof is deferred to another paper, and on two admitted inputs—the Ginibre density formula and the GUE-to-sine-kernel convergence—so if any of these fail, the chain from variance to spectrum loses support.

Editorial extensions

If this is right

  • To test hyperuniformity in practice, estimate the structure factor at small frequencies; Theorem 2.1 shows ball-variance, smooth-statistic variance, and S(B_ε)/ε^d are interchangeable.
  • Hyperuniformity is not tied to periodicity: any mixing process whose spectral density vanishes at the origin is hyperuniform, and in dimension 2, with extra integrability or finite Coulomb energy, such processes are L²-perturbed lattices.
  • The hyperuniformity exponent organizes the classification: class I (α>1), class II (α=1), class III (α<1); the GAF-zero process has exponent 4 and the Ginibre process exponent 2, explaining their differing rigidity levels.
  • Stealthy processes, whose spectrum vanishes in a neighborhood of the origin, are maximally rigid on strictly convex cones and have bounded holes, making them as close to crystals as a disordered process can be.
  • Rigidity of determinantal point processes is limited to number rigidity in dimensions 1 and 2; in higher dimensions the spectral density cannot vanish fast enough to allow higher-order rigidity.

Reading between the lines

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

  • The spectral characterization suggests a practical diagnostic: for finite samples, the scattering intensity at suitable low frequencies should vanish as the sample grows, so a model-free estimator based on tapered structure factors could provide a rigorous test of hyperuniformity.
  • If the rigidity criterion is taken at face value, the question of whether a disordered stealthy point process exists in dimension at least 2 becomes central: such a process would be simultaneously mixing and maximally rigid, a combination no current example achieves unless it is built from shifted lattices.
  • The transport results imply that in dimension 2 hyperuniformity plus finite Coulomb energy is a sufficient condition for a stationary point process to have a matching to the lattice with finite second moment; one could test numerically whether the condition is also necessary.
  • Extending the equivalence to non-Euclidean settings may require replacing balls by sets whose volume and boundary volume scale differently; the discussion of Gelfand spaces suggests a family of symmetric spaces where the same spectral framework could be applied.
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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

4 major / 5 minor

Summary. The manuscript is a survey/lecture-note treatment of hyperuniform random measures. It develops the framework of wide-sense stationary random measures, states and proves a spectral characterization of hyperuniformity (Theorem 2.1), and derives universal variance lower bounds (Theorem 2.2). It then applies the framework to emblematic examples: planar Gaussian analytic function zeros, Ginibre and GUE/Sine_β processes as determinantal point processes, perturbed lattices, and quasicrystalline cut-and-project models. The final chapters connect hyperuniformity to optimal transport / perturbed-lattice representations and to rigidity phenomena, including linear, number, and maximal rigidity. The central organizing claim is that second-order hyperuniformity—equivalently, vanishing of the spectral measure near zero—controls the large-scale number variance and, under additional conditions, leads to macroscopic order such as good transport properties and rigidity.

Significance. If its claims hold, this survey provides a useful unifying framework for a field that has grown rapidly across probability, statistical physics, and image analysis. The author is careful to separate what is proved in the text from what is imported from the literature, and several classical computations are worked out in full: the GOE change-of-variables Jacobian (Section 3.3.1), the GAF variance bound (Section 3.2.1), and the spectral measures of shifted and independently perturbed lattices (Section 2.1). The collection of recent results on transport, matching, and rigidity is valuable and mostly gives accurate pointers to the literature. The main weaknesses are that two load-bearing ingredients are quoted from the author's own unpublished/preprint material ([70]) or from an 'in preparation' paper ([71]), and one proof in the central Chapter 2 has a small but real gap. These issues are repairable and do not undermine the likely correctness of the results, but they need to be addressed before the survey can serve as a self-contained reference.

major comments (4)
  1. [§2.2, Lemma 2.2 and Theorem 2.1] The proof of Theorem 2.1(iii)⇒(i) relies on Lemma 2.2 to control the term R^{2d}∫_{B_1^c}(|u|R)^{-d-1}S(du), and Theorem 2.2 uses the same condition. Lemma 2.2 is stated in §2.1 as (2.4), but its proof is deferred to [70, Lemma 3], the author's own preprint. The precise exponent d+1 is load-bearing: if only a weaker tail bound held, the displayed estimates would not give o(R^d). I recommend either adding a proof of Lemma 2.2 in an appendix or citing a readily accessible published source, and adding a remark about what would change if the tail bound were weaker.
  2. [§2.3, proof of Theorem 2.2] The proof assumes that for some ρ0>0 one has S(B_{ρ0}^c)>0. This excludes, for example, M(dx)=Z dx with Var Z>0, whose spectral measure is a non-zero multiple of δ0. Such a measure satisfies the theorem's hypothesis 'not identically 0 a.s.' and the conclusion is true, but the displayed argument does not cover it. This is a repairable gap: the atom-at-zero case should be handled separately (the variance then grows like R^{2d}, so the lower bound is immediate), or the proof should be rephrased to avoid the assumption.
  3. [§5.2–5.3, Propositions 5.3 and Theorem 5.4] The claims of maximal rigidity for stealthy measures and for measures with purely atomic spectral density are stated as established results but are attributed to [71], 'in preparation'. Because Chapter 5 uses these claims as its main conclusions, the reader cannot currently verify them from the manuscript or from the published literature. I recommend either including proofs/sketches in an appendix, or explicitly marking these as forthcoming results and distinguishing them from established theorems.
  4. [§3.3.3, proof of Theorem 1.2] The proof that Ginibre eigenvalues form a DPP starts with 'We admit here the density representation (1.4)', and the GUE-to-sine-kernel convergence in §3.3.4 is left as a black box. For a survey this is acceptable if the reader is given precise statements with references, but the text should be explicit about which facts are imported. In particular, the density (1.4) is used to identify the DPP kernel and is load-bearing for the Ginibre example, so a proof or a specific reference to a complete proof is needed.
minor comments (5)
  1. [§5.1, first paragraph] The text refers to 'Proposition 2.4', which does not exist in the manuscript; the intended reference is presumably Proposition 2.1 or a similar variance-decay statement.
  2. [§2.5, paragraph before Theorem 2.3] Typo: 'Poison / sub-Poisson decay' should be 'Poisson / sub-Poisson decay'.
  3. [§3.5, Theorem 3.3] The first bullet writes 'lim_{ε→0} S(B_ε)/ε^d = op(ε^{d/d1})'. The left-hand side is a real number while the right-hand side is a stochastic-order symbol; this should be rewritten as an explicit rate, e.g. S(B_ε)/ε^d = O(ε^{d/d1}) or an asymptotic order statement.
  4. [§5.1.1, Remark 5.1] The monotonicity sentence contains a repeated symbol: 'k-rigidity for s1 implies k-rigidity for s1' is self-referential and should presumably read 'k-rigidity for s implies k-rigidity for s1'.
  5. [§4.2.2, Theorem 4.5] The statement 'if and only if d ≥ 3' should specify that this refers to the mean p-Wasserstein cost being O(n^d) for the listed i.i.d. uniform model; the current phrasing is slightly compressed for a survey and could mislead readers comparing with one-dimensional results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are either proved in-text or cited to independent sources; the self-citations do not reduce the claims to their own inputs.

full rationale

I walked the claimed derivation chain. Theorem 2.1's spectral characterization is proved in-text; the only deferred input is Lemma 2.2's spectral-tail bound, cited to the author's [70, Lemma 3]. This self-citation is load-bearing for the (iii)->(i) step, but it is not a reduction of the theorem to its own conclusion: the lemma is a general integrability statement about wide-sense stationary random measures, not equivalent to hyperuniformity, and the surrounding equivalence is an independent spectral calculation also credited to [14]. The rigidity and transport chapters rely mainly on independent sources ([50], [57], [32], [22], [113]) and on the author's [70] for additional results, but no fitted parameter is renamed as a prediction, no ansatz is smuggled in via self-citation, and no known result is merely redefined. The acknowledged omissions — the Ginibre density representation admitted in Section 3.3.3 and the GUE-to-sine-kernel convergence left as a black box in Section 3.3.4 — are missing proofs, not circular steps. The announced results in the in-preparation [71] are presented as future work, not as the basis of the earlier derivations. I therefore find no specific circular reduction of the kind required by the rules, and score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data anywhere; the hyperuniformity exponent α is a descriptor of a spectral measure, not a fitted constant. No new physical or mathematical entities (particles, forces, dimensions, constants) are postulated — Example 3.1's irrationally shifted lattices are a construction inside existing theory. The ledger instead records the survey's dependence on stationarity, on a self-cited spectral growth lemma, on admitted black boxes, and on unpublished work.

assumptions (6)
  • domain assumption Stationarity and L²_loc: the entire framework applies to wide-sense stationary L²_local random measures (Section 2.1); point-process theorems inherit this.
    All central results (Theorem 2.1, 2.2, 2.3, Chapter 5) are stated for this class. Non-stationary or edge-dominated systems are handled only via the separate asymptotic Definition 2.1 and Theorem 3.2.
  • standard math Lemma 2.2 spectral growth bound ∫(1+||u||)^{-d-1} S(du) < ∞, proof deferred to [70, Lemma 3].
    The proof of Theorem 2.1(iii)⇒(i) and Theorem 2.2 depend on this bound; it is quoted without proof from the author's own paper, so within this text it lacks independent verification.
  • domain assumption Ginibre density representation (1.4) is 'admitted' (Section 3.3.3); GUE-to-sine-kernel convergence is a black box (Section 3.3.4).
    These standard results underpin Theorem 1.2 and the hyperuniformity of Sine₂; the survey deliberately omits their proofs.
  • ad hoc to paper Maximal rigidity theorems (Proposition 5.3, Theorem 5.4) rely on [71], 'in preparation'.
    The survey presents these as established but the proof is in an unpublished manuscript by the same author, so this claim cannot be independently checked.
  • standard math Standard background theorems used without proof: Bochner's theorem, Plancherel identity, Poisson summation, Paley–Wiener theorem, Marcinkiewicz's theorem.
    Explicitly invoked at Sections 2.1–2.5 and 5.1.1; these are uncontroversial tools of harmonic analysis and probability.
  • domain assumption External existence results imported: Sineβ existence via Brownian carousel [113], Leblé's 2D Coulomb hyperuniformity (Theorem 3.2), AKT matching theorem [5].
    The survey's emblematic claims rest on these cited theorems, stated without re-derivation.

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Cite this review

Pith. "Pith review of Hyperuniform random measures, transport and rigidity." pith.science (2026). https://pith.science/paper/MKRNB522

@misc{pith2026251018392,
  author       = {Pith},
  title        = {Pith review of: Hyperuniform random measures, transport and rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKRNB522}},
  note         = {Machine review of arXiv:2510.18392}
}
read the original abstract

This survey explores the foundational theory and recent developments in the study of hyperuniformity. We present a comprehensive mathematical framework in the context of weakly stationary random measures, emphasizing spectral characterizations and second order asymptotics. Classical examples - including determinantal point processes, Gibbs measures, and zero sets of Gaussian analytic functions - are presented in depth to illustrate core principles. We also highlight recent progress connecting hyperuniformity with optimal transport and rigidity phenomena, pointing to emerging directions in the field.

Figures

Figures reproduced from arXiv: 2510.18392 by the authors.

Figure 1
Figure 1. (left) shows the photoreceptor locations of a bird’s eye, a class of species renowned for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1.2
Figure 1.2. Left. Disordered hyperuniform receptors [60]. Middle. Periodic “ordered” photoreceptors. Right. Dithering - Greyscale levels replaced by hyperuniform “blue noise” samples [30], ACM Trans. Graph. mathematicians and the reason why this field of study exists is that hyperuniformity is a very natural and universal way to mathematically define a certain form of regularity. Roughly speaking, a sample is hyperuniform if th… view at source ↗
Figure 1.3
Figure 1.3. Left. Ginibre ensemble. Right. GAF zeros [PITH_FULL_IMAGE:figures/full_fig_p013_1_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4.1
Figure 4.1. Figure 4.1: Sending kids to school. Left. Ginibre eigenvalues. Middle. Poisson points. Right. GAF zeros. Ilustration by D. Hawat 4.2.1 Optimal transport, matching and allocation We propose in this chapter a quantification of this concept in terms of matching and allocation. Let …

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Forward citations

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Reference graph

Works this paper leans on

117 extracted references · 1 canonical work pages · cited by 5 Pith papers

  1. [70]

    Lachièze-Rey

    R. Lachièze-Rey. Rigidity of random stationary measures and applications to point processes. https://arxiv.org/abs/2409.18519, 2024

    arXiv 2024

  2. [71]

    Lachièze-Rey

    R. Lachièze-Rey. The maximal rigidity phenomenon: stealthy systems, periodicity and qua- sicrystals. in preparation, 2025+

    2025

  3. [50]

    Ghosh and Y

    S. Ghosh and Y. Peres. Rigidity and tolerance in point processes: Gaussian zeros and ginibre eigenvalues. Duke Math. J., 166(10):1789–1858, 2017

    2017

  4. [93]

    Peres and A

    Y. Peres and A. Sly. Rigidity and tolerance for perturbed lattices. arXiv:1409.4490, 2014

    arXiv 2014

  5. [1]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun.Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. US Government printing office, 1968

    1968

  6. [2]

    Adhikari, S

    K. Adhikari, S. Ghosh, and J. Lebowitz. Fluctuation and entropy in spectrally constrained random fields. Comm. Math. Phys., 386:749–780, 2021

    2021

  7. [3]

    R. J. Adler and J. E. Taylor.Random Fields and Geometry. Springer, 2007

    2007

  8. [4]

    Aizenman and P

    M. Aizenman and P. A. Martin. Structure of Gibbs States of one Dimensional Coulomb Systems. Comm. Math. Phys., 78:99–116, 1980

    1980

Show all 117 references
  1. [5]

    Ajtai, J

    M. Ajtai, J. Komlos, and G. Tusnady. On optimal matchings.Combinatorica, 4:259–264, 1984

    1984

  2. [6]

    Anderson, A

    G. Anderson, A. Guionnet, and O. Zeitouni.An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 2010

    2010

  3. [7]

    Baake and U

    M. Baake and U. Grimm. Aperiodic order, vol. 1, Encyclopedia of Mathematics and its Applications, volume 149. Cambridge University Press, 2013

    2013

  4. [8]

    Baccelli, B

    F. Baccelli, B. Błaszczyszyn, and M. Karray. Random measures, point processes, and stochastic geometry, 2020

    2020

  5. [9]

    Bauerschmidt, P

    R. Bauerschmidt, P. Bourgade, M. Nikul, and H. Yau. Local density for two-dimensional one- component plasma. Comm. Math. Phys., 356:189–230, 2017

    2017

  6. [10]

    J. Beck. Irregularities of distribution. I.Acta Math., 159:1–49, 1987

    1987

  7. [11]

    Berg and G

    C. Berg and G. Frost.Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1975

    1975

  8. [12]

    M. V. Berry. Statistics of nodal lines and points in chaotic quantum billiards: perimeter cor- rections, fluctuations, curvature.Journal of Physics A: Mathematical and General, 35(13):3025, 2002

    2002

  9. [13]

    Björklund and M

    M. Björklund and M. Byléhn. Hyperuniformity and hyperfluctuations of random measures in commutative spaces.arXiv:2503.01567, 2025

    2025 arXiv

  10. [14]

    Hyperuniformityandnon-hyperuniformityofquasicrystals

    M.BjörklundandT.Hartnick. Hyperuniformityandnon-hyperuniformityofquasicrystals. Math. Ann., 389:365 – 426, 2024

    2024

  11. [15]

    Błaszczyszyn, D

    B. Błaszczyszyn, D. Yogeshwaran, and J. Yukich. Limit theory for geometric statistics of point processes having fast decay of correlations.Ann. Prob., 47:835–895, 2019

    2019

  12. [16]

    Bobkov and M

    S. Bobkov and M. Ledoux. A simple Fourier analytic proof of the AKT optimal matching theorem. Ann. Appl. Prob., 31(6):2567–2584, 2021. 59 R. Lachièze-Rey Hyperuniform random measures 60

    2021

  13. [17]

    A. I. Bufetov. Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel. Bull. Math. Sci., 6:163–172, 2016

    2016

  14. [18]

    A. I. Bufetov. Conditional Measures of Determinantal Point Processes. Funct. Anal. Appl., 54:7–20, 2020

    2020

  15. [19]

    A. I. Bufetov, Y. Dabrowski, and Y. Qiu. Linear rigidity of stationary stochastic processes. Ergod. Th. & Dynam. Sys., 38:2493–2507, 2018

    2018

  16. [20]

    A. I. Bufetov, P. P. Nikitin, and Y. Qiu. On number rigidity for Pfaffian point processes.Mosc. Math. J., 2:217–274, 2019

    2019

  17. [21]

    A. I. Bufetov and Y. Qiu. J-Hermitian determinantal point processes: balanced rigidity and balanced Palm equivalence.Math. Ann., 371:127–188, 2018

    2018

  18. [22]

    Butez, S

    R. Butez, S. Dallaporta, and D. Garcia-Zelada. On the Wasserstein distance between a hyper- uniform point process and its mean. https://arxiv.org/pdf/2404.09549.pdf, 2024

    2024 arXiv

  19. [23]

    Bylehn and M

    M. Bylehn and M. Bjorklund. Hyperuniformity of random measures on euclidean and hyperbolic spaces. arXiv:2405.12737, 2024

    2024 arXiv

  20. [24]

    Chatterjee

    S. Chatterjee. Rigidity of the three-dimensional hierarchical coulomb gas.Prob. Th. Rel. Fields, 175:1123–1176, 2019

    2019

  21. [25]

    Chatterjee, R

    S. Chatterjee, R. Peled, Y. Peres, and D. Romik. Gravitational allocation to Poisson points. Ann. Math., 172(1):617–671, 2010

    2010

  22. [26]

    Chhaibi and J

    R. Chhaibi and J. Najnudel. Rigidity of the Sineβ process. Elec. Comm. Prob., 94:1–8, 2018

    2018

  23. [27]

    I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai.Ergodic theory. Nauka, Moscow. English transl., Springer-Verlag, New York-Heidelberg-Berlin 1982., 1980

    1982

  24. [28]

    S. Coste. Order, fluctuations, rigidities. https://scoste.fr/assets/survey_hyperuniformity.pdf, 2021

    2021

  25. [29]

    D. J. Daley and D. Vere-Jones.An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods. Springer, Probability and its applications, 1988

    1988

  26. [30]

    de Goes, K

    F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun. Blue noise through optimal trans- port. ACM Trans. Graph., 31(6), 2012

    2012

  27. [31]

    Dereudre and D

    D. Dereudre and D. Flimmel. Non-hyperuniformity of Gibbs point processes with short-range interactions. J. Appl. Prob., 61(4):1380–1406, 2024

    2024

  28. [32]

    Dereudre, D

    D. Dereudre, D. Flimmel, T. Huessman, and T. Leblé. (Non)-hyperuniformity of perturbed lattices. https://arxiv.org/abs/2405.19881, 2024

    2024 arXiv

  29. [33]

    Dereudre, A

    D. Dereudre, A. Hardy, T. Leblé, and M. Maïda. DLR equations and rigidity for the sine-beta process. Comm. Pure Appl. Math., 74(1):172–222, 2020

    2020

  30. [34]

    Dereudre and T

    D. Dereudre and T. Vasseur. Number-rigidity andβ-circular Riesz gas.Ann. Prob., 51(3):1025– 1065, 2023

    2023

  31. [35]

    Dimitriu and A

    I. Dimitriu and A. Edelman. Matrix models for beta ensembles.J. Math. Phys., 43(11):5830– 5847, 2002

    2002

  32. [36]

    F. J. Dyson. Correlations between the eigenvalues of a random matrix.Comm. Math. Phys., 19(235-250), 1970. R. Lachièze-Rey Hyperuniform random measures 61

    1970

  33. [37]

    D. Flimmel. Fitting regular point patterns with a hyperuniform perturbed lattice. arXiv:2503.12179

    arXiv

  34. [38]

    P. J. Forrester. Log-gases and random matrices. London Mathematical Society Monographs. 2010

    2010

  35. [39]

    P. J. Forrester and G. Honner. Exact statistical properties of the zeros of complex random polynomials. J. Phys. A: Math. and General, 32(16):2961, 1999

    1999

  36. [40]

    Gabrielli, M

    A. Gabrielli, M. Joyce, and S. Torquato. Tilings of space and superhomogeneous point processes. Phys. Rev. E, 77:031125, 2008

    2008

  37. [41]

    Ganguly and S

    S. Ganguly and S. Sarkar. Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions.Prob. Th. Rel. Fields, 177:621–675, 2020

    2020

  38. [42]

    L. Gass. Spectral criteria for the asymptotics of local functionals of gaussian fields and their application to nodal volumes and critical. https://arxiv.org/pdf/2501.07356, 2025

    2025 arXiv

  39. [43]

    S. Ghosh. Determinantal processes and completeness of random exponentials: the critical case. Prob. Th. Rel. Fields, 163:643–665, 2015

    2015

  40. [44]

    S. Ghosh. Palm measures and rigidity phenomena in point processes. Elec. Comm. Prob., 21:1–14, 2016

    2016

  41. [45]

    Ghosh and M

    S. Ghosh and M. Krishnapur. Rigidity Hierarchy in Random Point Fields: Random Polynomials and Determinantal Processes.Comm. Math. Phys., 388:pp. 1205–1234, 2021

    2021

  42. [46]

    Ghosh, M

    S. Ghosh, M. Krishnapur, and Y. Peres. Continuum Percolation for Gaussian zeroes and ginibre eigenvalues. Ann. Prob., 44(5):3357–3384, 2016

    2016

  43. [47]

    Ghosh and J

    S. Ghosh and J. Lebowitz. Number rigidity in superhomogeneous random point fields.J. Stat. Phys., 166(3-4), 2017

    2017

  44. [48]

    Ghosh and J

    S. Ghosh and J. L. Lebowitz. Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey.Indian J. of Pure and Appl. Math., 48(4):609–631, 2017

    2017

  45. [49]

    Ghosh and J

    S. Ghosh and J. L. Lebowitz. Generalized stealthy hyperuniform processes: Maximal rigidity and the bounded holes conjecture.Comm. Math. Phys., 363:97–110, 2018

    2018

  46. [51]

    Haimi1, G

    A. Haimi1, G. Koliander, and J. L. Romero. Zeros of gaussian weyl–heisenberg functions and hyperuniformity of charge.J. Stat. Phys., 187(22):1–41, 2022

    2022

  47. [52]

    Hawat, G

    D. Hawat, G. Gautier, R. Bardenet, and R. Lachièze-Rey. On estimating the structure factor of a point process, with applications to hyperuniformity.Statistics and computing, 33(61), 2023

    2023

  48. [53]

    Hoffman, A

    C. Hoffman, A. E. Holroyd, and Y. Peres. A stable marriage of Poisson and Lebesgue.Annals of Probability, 34(4):1241–1272, 2006

    2006

  49. [54]

    A. E. Holroyd and T. Soo. Insertion and deletion tolerance of point processes. Electron. J. Probab, 18(74):DOI: 10.1214/EJP.v18–2621, 2013

    2013 doi

  50. [55]

    Holroyd, R

    E. Holroyd, R. Pemantle, Y. Peres, and O. Schramm. Poisson matching.Ann. IHP Prob. Stat., 45(1):266–287, 2009. R. Lachièze-Rey Hyperuniform random measures 62

    2009

  51. [56]

    J. B. Hough, M. Krishnapur, Y. Peres, and B. Viràg.Zeros of Gaussian Analytic Functionsand Determinantal Point Processes. University Lecture Series. Institute of Mathematical Statistics, 2009

    2009

  52. [57]

    Thelinkbetweenhyperuniformity, Coulombenergy, andWasserstein distance to Lebesgue for two-dimensional point processes

    M.HuesmannandT.Leblé. Thelinkbetweenhyperuniformity, Coulombenergy, andWasserstein distance to Lebesgue for two-dimensional point processes. https://arxiv.org/abs/2404.18588, 2024

    2024

  53. [58]

    J. Jalowy. The Wasserstein distance to the circular law.Annales de l’Institut Henri Poincare (B) Probabilites et statistiques, 59(4):2285–2307, 2023

    2023

  54. [59]

    Jalowy and H

    J. Jalowy and H. Stange. Box-covariances of hyperuniform point processes. arXiv:2506.13661, 2025

    2025 arXiv

  55. [60]

    Y. Jiao, T. Lau, H. Hatzikirou, M. Meyer-Hermann, J. C. Corbo, and S. Torquato. Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem. Phys. Rev. E, 89(022721), 2014

    2014

  56. [61]

    Klatt, G

    M. Klatt, G. Last, L. Lotz, and D. Yogeshwaran. Invariant transports of stationary random mea- sures: asymptotic variance, hyperuniformity, and examples. https://arxiv.org/abs/2506.05907, 2025

    2025 arXiv

  57. [62]

    M. A. Klatt and G. Last. On strongly rigid hyperfluctuating random measures.J. Appl. Prob., 59(4):948–961, 2022. https://arxiv.org/abs/2008.10907

    2022 arXiv

  58. [63]

    M. A. Klatt, G. Last, and D. Yogeshwaran. Hyperuniform and rigid stable matchings.Rand. Struct. Alg., 57:439–473, 2020

    2020

  59. [64]

    M. A. Klatt, J. Lovric, D. Chen, S. Kapfer, F. Schaller, P. Schonoffer, B. Gardiner, A. Smith, G. Schroder-Turk, and S. Torquato. Universal hidden order in amorphous cellular geometries. Nature communications, 10(811), 2019

    2019

  60. [65]

    A. N. Kolmogorov. Stationary sequences in Hilbert space.Bull. Moskov.Gos. Univ. Mat., 2:1–40, 1941

    1941

  61. [66]

    Krishnapur and B

    M. Krishnapur and B. Virág. The ginibre ensemble and gaussian analytic functions.Int. Math. Res. Not., 6:1441–1464, 2014

    2014

  62. [67]

    Krishnapur and D

    M. Krishnapur and D. Yogeshwaran. Stationary random measures: Covariance asymptotics, variance bounds and central limit theorems. arXiv:2411.08848, 2024

    2024 arXiv

  63. [68]

    Lachièze-Rey

    R. Lachièze-Rey. Diophantine Gaussian excursions and random walks. https://arxiv.org/abs/2104.07290

    arXiv

  64. [69]

    Lachièze-Rey

    R. Lachièze-Rey. Variance linearity for real Gaussian zeros.Ann. I. H. Poincarré B, 58(4), 2022

    2022

  65. [72]

    Lachièze-Rey and D

    R. Lachièze-Rey and D. Yogeshwaran. Hyperuniformity and optimal transport of point processes. arXiv, 2024

    2024

  66. [73]

    P. Y. G. Lamarre, P. Ghosal, and Y. Liao. Spectral rigidity of random Schrödinger operator via Feynman-Kac formulas. Ann. I. H. Poincarré B, 21:2259–2299, 2020. R. Lachièze-Rey Hyperuniform random measures 63

    2020

  67. [74]

    T. Leblé. DLR equations, number-rigidity and translation-invariance for infinite-volume limit points of the 2DOCP. arXiv:2410.04958

    arXiv

  68. [75]

    T. Leblé. The two-dimensional one-component plasma is hyperuniform. arXiv, 2023

    2023

  69. [76]

    Lebowitz

    J. Lebowitz. Charge fluctuations in coulomb systems.Phys. Rev. A, 27(3):1491, 1983

    1983

  70. [77]

    M. Lewin. Coulomb and Riesz gases: The known and the unknown.J.Math.Phys., 63(6):061101, https://doi.org/10.1063/5.0086835 2022

    2022 doi

  71. [78]

    Lyons and J

    R. Lyons and J. E. Steif. Stationary Determinantal Processes: Phase Multiplicity, Bernoullicity, Entropy, and Domination.Duke Math. J., 120(3):515–575, 2003

    2003

  72. [79]

    J. K. M. A. Klatt and S. Torquato. Cloaking the underlying long-range order of randomly perturbed lattices. Phys. Rev. E, 101(032118), 2020

    2020

  73. [80]

    O. Macchi. The coincidence approach to stochastic processes.Adv. Appl. Prob., 7:83–122, 1975

    1975

  74. [81]

    P. A. Martin and T. Yalcin. The charge fluctuations in classical Coulomb systems.Journal of Statistical Physics, 22:435–463, 1980

    1980

  75. [82]

    Mastrilli, B

    G. Mastrilli, B. Blaszczyszyn, and F. Lavancier. Estimating the hyperuniformity exponent of point processes. arXiv:2407.16797, 2024

    2024 arXiv

  76. [83]

    M. L. Mehta. Pure and applied mathematics. InRandom Matrices, volume 142. Academic Press, Elsevier, 2004

    2004

  77. [84]

    Mérigot, F

    Q. Mérigot, F. Santambrogio, and C. Sarrazin. Non-asymptotic convergence bounds for wasser- stein approximation using point clouds.Adv. NeurIPS, 34, 2021

    2021

  78. [85]

    Molchanov.Theory of random sets

    I. Molchanov.Theory of random sets. Springer-Verlag, London, 2005

    2005

  79. [86]

    P. K. Morse, J. Kim, P. J. Steinhardt, and S. Torquato. Generating large disordered stealthy hyperuniform systems with ultrahigh accuracy to determine their physical properties.Phys.Rev. Res., 5(3):33190, 2023

    2023

  80. [87]

    Nazarov and M

    F. Nazarov and M. Sodin. Correlation functions for random complex zeroes: strong clustering and local universality.Communications in Mathematical Physics, 310(1):75–98, 2012

    2012

  81. [88]

    Nazarov, M

    F. Nazarov, M. Sodin, and A. Volberg. Transportation to random zeroes by the gradient flow. Geom. Funct. Anal, 17:887–935, 2007

    2007

  82. [89]

    Nourdin, G

    I. Nourdin, G. Peccati, and M. Rossi. Nodal Statistics of Planar Random Waves.Comm. Math. Phys., 369:99–151, 2019

    2019

  83. [90]

    E. G. Oguz, J. E. Socolar, P. J. Steinhardt, and S. Torquato. Hyperuniformity and anti- hyperuniformity in onedimensional substitution tilings. Acta Crystallographica Section A: Foundations and Advances, 75(1):3–13, 2019

    2019

  84. [91]

    H. Osada. Vanishing self-diffusivity in Ginibre interacting Brownian motions in two dimensions. Prob. Th. Rel. Fields, ttps://doi.org/10.1007/s00440-024-01303-2, 2024

    2024 doi

  85. [92]

    Parnovski and A

    L. Parnovski and A. V. Sobolev. On the bethe-sommerfeld conjecture for the polyharmonic operator. Duke Math. J., 107(2):209–238, 2001

    2001

  86. [94]

    Peres and B

    Y. Peres and B. Virág. Zeros of the i.i.d. gaussian power series: a confor- mally invariant determinantal process. Acta Math., 194:1–35, 2005. R. Lachièze-Rey Hyperuniform random measures 64

    2005

  87. [95]

    Pilleboue, G

    A. Pilleboue, G. Singh, D. Coeurjolly, M. Kazhdan, and V. Ostromoukhov. Variance analysis for monte carlo integration.ACM Trans. Graph., 34(4), 2015

    2015

  88. [96]

    Prod’Homme.Contributionsto the optimal transport problem and its regularity

    M. Prod’Homme.Contributionsto the optimal transport problem and its regularity. PhD thesis, Université Paul Sabatier-Toulouse III, 2021.https://theses.hal.science/tel-03419872/

    2021

  89. [97]

    W. Rudin. Functional Analysis. McGraw-Hill, Inc., 1991

    1991

  90. [98]

    Santambrogio.Optimal Transport for Applied Mathematicians

    F. Santambrogio.Optimal Transport for Applied Mathematicians. Birkhaüser, Basel, 2015

    2015

  91. [99]

    S. Serfaty. Systems of points with Coulomb interactions.Proc. ICM 2018, pages 935–977, 2019

    2018

  92. [100]

    A. Shih, M. Casiulis, and S. Martiniani. Fast generation of spectrally shaped disorder.Phys. Rev. E, 110(034122), 2024

    2024

  93. [101]

    M. M. Skryganov. Constructions of uniform distributions in terms of geometry of numbers. Algebra i Analiz, 6(3):200–230, 1994

    1994

  94. [102]

    M. Sodin. Zeros of gaussian analytic functions.Math. Res. Lett., 7(4):371–381, 2000

    2000

  95. [103]

    Sodin and B

    M. Sodin and B. Tsirelson. Random complex zeroes I. Asymptotic normality. Isr. J. Math., 144:125–149, 2004

    2004

  96. [104]

    Sodin, A

    M. Sodin, A. Wennman, and O. Yakir. The random weierstrass zeta function ii. fluctuations of the electric flux through rectifiable curves.J. Stat. Phys, 190(164), 2023

    2023

  97. [105]

    Soshnikov

    A. Soshnikov. Gaussianl imit for determinantal random point fields.Ann. Prob., 30:171–187, 2002

    2002

  98. [106]

    G. Szegö. Beitrage zur Theorie der Toeplitzschen Formen.Math. Z., 6:167–202, 1921

    1921

  99. [107]

    Torquato

    S. Torquato. Disordered hyperuniform heterogeneous materials. J. Phys.: Condens. Matter, 28(414012), 2016

    2016

  100. [108]

    Torquato

    S. Torquato. Hyperuniformity and its generalizations.Phys. Rev. E, 94(022122), 2016

    2016

  101. [109]

    Torquato

    S. Torquato. Hyperuniform States of Matter.Physics Reports, 745:1–95, 2018

    2018

  102. [110]

    Torquato and F

    S. Torquato and F. H. Stillinger. Local Density Fluctuations, Hyperuniform Systems, and Order Metrics. Phys. Rev. E, 68(041113):1–25, 2003

    2003

  103. [111]

    Torquato, G

    S. Torquato, G. Zhang, and F. H. Stillinger. Ensemble theory for stealthy hyperuniform disor- dered ground states.Phys. Rev. X, 5(021020):1–23, 2015

    2015

  104. [112]

    R. A. Ullichney. Dithering with blue noise.Proc. IEEE, 76(1):56–79, 1988

    1988

  105. [113]

    Valkó and B

    B. Valkó and B. Viràg. Continuum limits of random matrices and the Brownian carousel.Invent. math., 177(3):463–508, 2009

    2009

  106. [114]

    C. Villani. Topics in optimal transportation. Graduate Studies in Mathematics, Vol. 58, 2003

    2003

  107. [115]

    D. Yan, J. Guo, B. Wang, X. Zhang, and P. Wonka. A Survey of Blue-Noise Sampling and Its Applications. J. Comput. Sci. Technol., 30:439–452, 2015

    2015

  108. [116]

    Zhang, F

    G. Zhang, F. Stillinger, and S. Torquato. Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations.Phys. Rev. E, 92 - 022119(1-14), 2015

    2015

  109. [117]

    Zhang, F

    G. Zhang, F. H. Stillinger, and S. Torquato. Can exotic disordered ”stealthy” particle configu- rations tolerate arbitrarily large holes?Soft matter, 36(https://doi.org/10.1039/C7SM01028A), 2017

    2017 doi

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