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REVIEW 3 major objections 3 minor 137 references

A Shared Observation Shields Collective Fluctuations while Preserving Local Independence

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Conditioning a population on its own record imposes a centered-square penalty and a nonpositive shield; measured heterogeneity is the excess over a negative baseline.

desk verdict Clean mathematical core and a nice conceptual point, but the advertised chi_4 baseline is a scope gap the paper itself acknowledges. read the letter →

arxiv 2608.08358 v1 pith:5UHLWSRL submitted 2026-08-08 cond-mat.stat-mech physics.optics

Hu Cang This is my paper · ORCID
classification cond-mat.stat-mechphysics.optics MSC 60H1060G4482C31 PACS 05.40.-a64.70.P
keywords dynamicalheterogeneityfour-pointsusceptibilityGirsanovtransformationpropagationofchaosSchurshieldglasstransitionconditionedensembles
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

This paper proves that conditioning a population of stochastic trajectories on a measured record that the population itself generated—the guiding example is a tagged tracer and the $z$ caging neighbors whose collective force drives its recorded history—multiplies the independent joint law of the hidden paths by exactly one term, a centered-square penalty along the single collective direction the record can see. Local independence survives this conditioning: any fixed pair keeps covariance of order $1/z$ and mutual information of order $1/z^2$, yet the $z(z-1)$ weak correlations add coherently into a finite, nonpositive suppression of collective fluctuations, an operator inequality the paper calls the Schur shield. Because the conditioning-only contribution is nonpositive, the paper concludes that a measured four-point susceptibility $\chi_4$ must be read as genuine cooperative signal plus a computable negative baseline: the true signal is the excess of the measurement over that baseline, not over zero. This reframes a long-standing interpretive question in glass physics, because growth of dynamical heterogeneity approaching arrest cannot originate in the conditioned sector and must come from genuine correlations that beat the shield. An exactly solvable Brownian model calibrates the construction, and the sign threshold between the negative shield and positive history-to-history propensity variance gives a quantitative test that existing simulation data can already perform.

What carries the argument

The load-bearing object is the conditional posterior of Eq. (2), a single centered-square penalty produced by the Girsanov transformation: it is rank-one, acts only along the collective direction the record can see, and carries the minus sign that makes the correction a suppression. Diagonalizing the one-path covariance operator $C$ turns the fixed-history collective covariance into $D = aC(aI+C)^{-1}$, so $D - C = -C^2(aI+C)^{-1} \preceq 0$, the operator inequality the paper names the Schur shield; each eigenmode of variance $c$ is suppressed by the factor $a/(a+c)$, weakly visible modes barely change, and strongly visible modes saturate at the observation-noise scale. A leave-two-out expansion produces the $O(z^{-1})$ pair covariance of Eq. (6) with mutual information $O(z^{-2})$, the resummation of those weak pair terms yields the parameter-free negative correction of Eq. (8), and the law of total variance splits the measured cross-correlation into the negative mean shield $-E\delta_X$ and the positive propensity variance $\mathrm{Var}(h_z(X))$, giving the sign threshold of Eq. (11).

What would settle it

In the exactly solvable Ornstein–Uhlenbeck model of Eq. (12), measure the fixed-history cross-covariance of two hidden paths and check the predicted closed form $\mathrm{Cov}[y_i(t), y_j(s)\,|\,x] = -(\sigma^2/z)[e^{-\lambda|t-s|} - (\lambda/\alpha)e^{-\alpha|t-s|}]$: a positive or order-one value at fixed history would falsify the Schur-shield claim. In any conditioned ensemble, the three quantities in Eq. (11) must satisfy $z\,\mathrm{Cov}(u_1,u_2) = \mathrm{Var}(h_z(X)) - E\delta_X$, so a systematic violation would expose residual interactions beyond the single centered-square penalty.

Watch

Extended reading notes

Core claim

The central discovery is a closed-form conditional posterior for a population that drives its own observation: with the record $X$ fixed, the density of the $z$ hidden trajectories with respect to the independent reference law is $dP_x^z / dQ^{\otimes z}_x = (1/Z_z(x)) \exp[-(1/(2az)) \sum_i (G_x(Y^i) - m_x)^2]$, Eq. (2), obtained by a Girsanov change of measure. From this single centered square the paper derives the fixed-history collective covariance $D = aC(aI+C)^{-1}$ and the Schur shield $D - C = -C^2(aI+C)^{-1} \preceq 0$, Eqs. (4)-(5); the $O(z^{-1})$ pair covariance and the resummed order-one correction $-\langle b_u, (aI+C)^{-1} b_u \rangle$, Eqs. (6)-(8); and, in the fluctuating-history ensemble, the identity $z\,\mathrm{Cov}(u_1,u_2) = \mathrm{Var}(h_z(X)) - E\delta_X$, Eq. (11), which sets the sign of measured cross-particle correlation by competition between the negative shield and the positive propensity variance. The paper's physical conclusion is that measured dynamical heterogeneity contains a conditioning-only component that is computable and nonpositive, so the genuine cooperative signal is the excess of the measured $\chi_4$ over this negative baseline, not over zero.

Load-bearing premise

The theorem assumes that, once the observed history is fixed, the $z$ hidden trajectories are independent apart from one collective drift term, so every pair correlation entering a measured $\chi_4$ is attributed to the observation itself rather than to direct neighbor-neighbor interactions or correlations between different tagged populations.

Editorial extensions

If this is right

  • Any four-point susceptibility computed from trajectories that generated the record it summarizes inherits a computable, nonpositive conditioning-only baseline; genuine dynamical correlations are the excess over that baseline, and comparing with zero underestimates them by exactly the mean shield $E\delta_X$.
  • Pair independence does not bound the collective response: fixed labels decouple as $O(z^{-1})$ while the resummed susceptibility keeps an order-one correction, so small mutual information of order $O(z^{-2})$ cannot rule out a coherent collective effect.
  • The sign of a measured cross-particle correlation is protocol-dependent: experiments that fix the macroscopic history weight the negative shield, while experiments that mix histories add the positive propensity variance, so two protocols can report opposite signs while agreeing on the underlying dynamics.
  • Tuning the output-noise variance $a$ gives a direct experimental handle: output-visible collective modes are suppressed by the factor $a/(a+c)$ while orthogonal observables are unaffected, and the exactly solvable model shows that equal-time and time-integrated responses differ, so no single static coupling reproduces both.
  • The exact theorem covers one tagged population; carrying the baseline to the bulk $\chi_4$, which compares fluctuations across different tagged stars, requires controlling cross-population correlations and is stated by the paper as a concrete open problem.

Reading between the lines

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

  • A reporting standard follows that the paper sketches but does not name: laboratories and simulations that store many histories could report the fixed-history shield and the between-history propensity variance separately, and Eq. (11) then decides the sign of genuine cooperativity without any new experiment.
  • The rank-one structure suggests a diagnostic that extends beyond glasses: for any collective observable built from the trajectories it describes, sweeping the measurement noise should suppress only the component overlapping the recorded mode, cleanly separating observation-induced shielding from intrinsic correlations in crowded-media probes, coarse-grained order parameters, or pooled neural recor
  • Because the reference law assumes hidden paths are independent given the record, the identity of Eq. (11) can be run in reverse: a systematic violation of $z\,\mathrm{Cov}(u_1,u_2) = \mathrm{Var}(h_z(X)) - E\delta_X$ in an interacting system would quantify the direct neighbor-neighbor and cross-population correlations that the observation-only construction omits.
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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

3 major / 3 minor

Summary. The paper analyzes how conditioning on a shared, endogenously generated observation affects the collective fluctuations of a population of hidden stochastic paths. In the finite-population model of Sec. II, the author derives, via a Girsanov transformation, a closed-form conditional posterior (Eq. (2)) in which the many-body effect is a single centered-square penalty along the record-visible collective direction. From this, Eq. (4)-(5) give the fixed-history collective covariance and the nonpositive 'Schur shield' correction. Section III derives pair-level independence with O(z^{-1}) covariance and O(z^{-2}) mutual information while showing that the resummed collective correction is order one. Section IV lets the observed history fluctuate and obtains the law-of-total-variance decomposition Eq. (11), separating a negative fixed-history shield from a positive propensity-variance term. Section V evaluates the construction exactly in a Gaussian model, and Sec. VI proposes a measurement workflow and a baseline interpretation for dynamical heterogeneity: measured chi_4 should be read as genuine cooperative signal plus a nonpositive conditioning-only baseline.

Significance. If the advertised scope were fully established, the paper would provide a useful exact reference calculation for observation-induced collective effects in a broad class of stochastic systems, and it would sharpen the interpretation of four-point susceptibility measurements in glassy liquids. The mathematical core is attractive: Eq. (2) is a clean and seemingly correct Girsanov result under the stated boundedness and Lipschitz assumptions, Eq. (5) is an algebraic identity, the operator formulas are parameter-free, and the Gaussian calibration in Sec. V and Appendix E gives an explicit falsifiable prediction for the equal-time versus time-integrated response. The paper is also refreshingly honest about some limitations, explicitly flagging the lack of uniformity on growing time windows and the open bulk-chi_4 extension. However, the load-bearing steps leading from the exact theorem to the advertised physical baseline currently rest on an unproven remainder estimate and on a scope extension that the paper itself declares open; these gaps need to be closed before the central physical claim is fully supported.

major comments (3)
  1. [Appendix B, Eq. (6)] The leave-two-out expansion in Eq. (B2) is presented as a sketch: the O_{L1}(z^{-3/2}) remainder is asserted but no explicit bound is derived, and no uniformity over the observables u and v is stated. Since Sec. II promises that 'all remainders quoted below are nonasymptotic and dimension free,' this is a missing load-bearing justification for the O(z^{-1}) covariance formula and for the resummation leading to Eq. (8). Please supply explicit remainder estimates, or explicitly downgrade the finite-z statements to formal asymptotics.
  2. [Sec. VI versus Abstract, Eq. (11)] The baseline subtraction in Eq. (11) is derived for a single tagged population under the independent-neighbor reference law of Sec. II and Appendix A. Section VI correctly states that carrying the theorem to bulk chi_4, which compares fluctuations across different tagged stars, requires controlling cross-population correlations and remains an open problem. The Abstract nevertheless says that 'a comparison that existing simulation data can already perform' is available. This is a scope mismatch: without a specified cross-population extension or an explicit matching of Eq. (11) to a particular bulk chi_4 protocol, existing simulation data cannot be used to test the advertised baseline. Please restrict the claim to the tagged-population susceptibility or provide the missing extension/protocol.
  3. [Sec. VI, Appendix F] The physical conclusion that the genuine cooperative signal is the measured value minus the negative baseline relies on Eq. (11) being applicable to real glassy systems, but the manuscript contains no simulation or data analysis check. Appendix F is a proposed workflow, not a validation. Given the Abstract's claim that existing simulation data can already perform the comparison, I would expect at least one concrete demonstration (for example, a reanalysis of isoconfigurational-ensemble simulations, or a numerical test of Eqs. (10)-(11) on a model glass former). Absent that, the manuscript should clearly state that the baseline interpretation is an application proposal rather than an established empirical result.
minor comments (3)
  1. [Sec. IV, first paragraph] In the sentence 'Let X denote the sigma algebra generated by the observed history,' the symbol X is reused for both the path and the sigma algebra; this is understandable but worth disambiguating, for instance by writing sigma(X).
  2. [Eq. (13)] The conditional covariance in Eq. (13) is stated for the complete stationary x history; the notation Cov[yi(t), yj(s)|x] could be misread as conditioning on the initial value. Please write explicitly 'conditional on the full sample path X = x' at that point.
  3. [Fig. 2 caption] The caption says 'One shared output constrains a whole population' but the figure itself is schematic; please state explicitly that the diagram is illustrative and not a plot of a computed quantity, to avoid confusion with the quantitative panels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is a self-contained exact conditioning identity with no fitted inputs, and the chi4 baseline is an exact decomposition with explicitly stated scope limits.

full rationale

The paper's derivation chain is self-contained and non-circular. Equation (2) is obtained by a Girsanov change of measure applied to the explicit model of Section II and Appendix A, with all assumptions stated (bounded f, Lipschitz coefficients, fixed window). The Schur shield D = aC(aI+C)^(-1) and the inequality D - C <= 0 in Eqs. (4)-(5) follow from Gaussian operator algebra, not from any fitted parameter. Equations (6) and (8) are leave-two-out expansions and exact resummations with stated remainders. Equation (11) is an identity from the law of total variance applied to exchangeable labels, and the proposed baseline subtraction (measured zCov plus E delta_X) is a rearrangement of that identity, not a statistical fit disguised as a prediction. The paper explicitly flags its scope limits: Section II notes the nonasymptotic constants are not uniform on windows growing with relaxation time, Section V repeats that a mixing estimate would be needed for such windows, and Section VI states that carrying the theorem to bulk chi4 is an open problem requiring control of cross-population correlations. These are honest scope statements, not circular reasoning. The only self-citations (Refs. 21-24) are historical context for time-domain optical spectroscopy and play no load-bearing role in the derivation. No fitted parameter is later renamed as a prediction, and no uniqueness claim is imported from the authors' prior work to force a choice. Thus there is no circular step by the standards of this review.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central calculation is self-contained given Girsanov and standard covariance conditioning. No numerical constant is fitted to data; the Gaussian model in Sec. V is a solvable illustration. The physical application imports the independent-paths-shared-record model as an assumption about how glassy dynamics factorizes, and that assumption is not derived from a specific liquid Hamiltonian.

assumptions (4)
  • standard math Girsanov theorem and Novikov condition for the change of measure from reference to endogenous dynamics
    Invoked in Sec. II and Appendix A to obtain the density Eq. (A2); requires f bounded and Lipschitz coefficients.
  • domain assumption Hidden paths Y^1,...,Y^z are independent given the observed history X under the reference law, and exchangeable
    Stated in Sec. II and Appendix A; all pair correlation in the model is generated by conditioning on X, not by direct neighbor-neighbor forces.
  • domain assumption The record sees only the additive, z^{-1/2}-normalized collective drift f(X_t, Y^i_t), and the full path X over [0,T] is observed
    Eq. (1) and Appendix A; the exact centered-square form and the z^{-1} pair covariance scale depend on this normalization and on full-path observation.
  • domain assumption A measured four-point susceptibility or dynamic propensity can be matched to this conditional ensemble with matching history, normalization, and fixed variables
    Secs. IV and VI assert that Eq. (11) supplies a baseline for chi_4; Appendix D itself says ensembles must be matched before comparing amplitudes.

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Cite this review

Pith. "Pith review of A Shared Observation Shields Collective Fluctuations while Preserving Local Independence." pith.science (2026). https://pith.science/paper/5UHLWSRL

@misc{pith2026260808358,
  author       = {Pith},
  title        = {Pith review of: A Shared Observation Shields Collective Fluctuations while Preserving Local Independence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5UHLWSRL}},
  note         = {Machine review of arXiv:2608.08358}
}
abstract

As a liquid approaches its glass transition, its dynamics turns heterogeneous: mobile and immobile regions coexist, and the four-point susceptibility $\chi_4$ that quantifies this heterogeneity grows sharply. Interpreting that growth is subtle, because the collective signals experiments record, such as a tagged particle's trajectory, an overlap function, or a mean field, are generated by the same particles they describe. Here we compute exactly what conditioning on such a shared record does to the population that produced it, for a broad class of stochastically observed systems; the guiding example is a tagged particle and the cage of $z$ neighbors that drives its force history. Using a Girsanov path transformation, we prove that the conditioning multiplies the independent joint law of the $z$ trajectories by exactly one term: a centered-square penalty along the single collective direction the record can see. Any fixed pair of particles stays nearly independent, with covariance falling as $O(z^{-1})$ and mutual information as $O(z^{-2})$, the property known as propagation of chaos, yet the $z(z-1)$ weak pair correlations add coherently into a finite suppression of collective fluctuations, the Schur shield $D - C = -C^2(aI + C)^{-1} \preceq 0$. An exactly solvable Brownian model calibrates the construction. The physical consequence is a calculable baseline for dynamical heterogeneity: conditioning itself contributes a computable, nonpositive amount to the susceptibility of a conditioned ensemble, so the genuine cooperative signal is the excess of the measured $\chi_4$ over this baseline rather than over zero, a comparison that existing simulation data can already perform.

Figures

Figures reproduced from arXiv: 2608.08358 by the authors.

Figure 1
Figure 1. FIG. 1. One shared output constrains a whole population. The diagram is schematic and has no coordinate axes. In the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A shared output selectively absorbs collective motion. (a) An eigenmode of one-path variance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Two sources set the sign of the cross-particle response. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

Works this paper leans on

137 extracted references · 60 canonical work pages

  1. [1]

    author author C. A. \ Angell ,\ https://doi.org/10.1126/science.267.5206.1924 journal journal Science \ volume 267 ,\ pages 1924 ( year 1995 ) NoStop

    arXiv 1924

  2. [2]

    author author M. D. \ Ediger ,\ https://doi.org/10.1146/annurev.physchem.51.1.99 journal journal Annual Review of Physical Chemistry \ volume 51 ,\ pages 99 ( year 2000 ) NoStop

    doi:10.1146/annurev.physchem.51.1.99

  3. [3]

    author author P. G. \ Debenedetti \ and\ author F. H. \ Stillinger ,\ https://doi.org/10.1038/35065704 journal journal Nature \ volume 410 ,\ pages 259 ( year 2001 ) NoStop

    doi:10.1038/35065704 2001

  4. [4]

    Berthier \ and\ author G

    author author L. Berthier \ and\ author G. Biroli ,\ https://doi.org/10.1103/RevModPhys.83.587 journal journal Reviews of Modern Physics \ volume 83 ,\ pages 587 ( year 2011 ) NoStop

    doi:10.1103/revmodphys.83.587 2011

  5. [5]

    Berthier , author G

    author author L. Berthier , author G. Biroli , author J.-P. \ Bouchaud , author L. Cipelletti , author D. El Masri , author D. L'H \^o te , author F. Ladieu ,\ and\ author M. Pierno ,\ https://doi.org/10.1126/science.1120714 journal journal Science \ volume 310 ,\ pages 1797 ( year 2005 ) NoStop

    doi:10.1126/science.1120714

  6. [6]

    Widmer-Cooper , author P

    author author A. Widmer-Cooper , author P. Harrowell ,\ and\ author H. Fynewever ,\ https://doi.org/10.1103/PhysRevLett.93.135701 journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 135701 ( year 2004 ) NoStop

    doi:10.1103/physrevlett.93.135701 2004

  7. [8]

    Tong \ and\ author H

    author author H. Tong \ and\ author H. Tanaka ,\ https://doi.org/10.1103/PhysRevX.8.011041 journal journal Phys. Rev. X \ volume 8 ,\ pages 011041 ( year 2018 ) NoStop

    work page doi:10.1103/physrevx.8.011041 2018

  8. [9]

    Ben Arous \ and\ author O

    author author G. Ben Arous \ and\ author O. Zeitouni ,\ https://doi.org/10.1016/S0246-0203(99)80006-5 journal journal Ann. Inst. H. Poincar \'e Probab. Statist. \ volume 35 ,\ pages 85 ( year 1999 ) NoStop

    work page doi:10.1016/s0246-0203(99)80006-5 1999

Show all 137 references
  1. [12]

    Kac ,\ in\ @noop booktitle Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability ,\ Vol

    author author M. Kac ,\ in\ @noop booktitle Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability ,\ Vol. volume 3 \ ( publisher University of California Press ,\ address Berkeley ,\ year 1956 )\ pp.\ pages 171--197 NoStop

    1956

  2. [13]

    author author H. P. \ McKean ,\ in\ @noop booktitle Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7) \ ( publisher Catholic University ,\ address Washington, D.C. ,\ year 1967 )\ pp.\ pages 41--57 NoStop

    1967

  3. [14]

    author author R. S. \ Ellis \ and\ author C. M. \ Newman ,\ https://doi.org/10.1007/BF00533049 journal journal Z. Wahrscheinlichkeitstheorie verw. Gebiete \ volume 44 ,\ pages 117 ( year 1978 ) NoStop

    1978 doi

  4. [15]

    Bolthausen ,\ https://doi.org/10.1007/BF00699109 journal journal Z

    author author E. Bolthausen ,\ https://doi.org/10.1007/BF00699109 journal journal Z. Wahrscheinlichkeitstheorie verw. Gebiete \ volume 72 ,\ pages 305 ( year 1986 ) NoStop

    1986 doi

  5. [16]

    Kusuoka \ and\ author S

    author author S. Kusuoka \ and\ author S. Liang ,\ https://doi.org/10.1007/PL00008727 journal journal Probab. Theory Relat. Fields \ volume 116 ,\ pages 221 ( year 2000 ) NoStop

    doi

  6. [17]

    Bally \ and\ author E

    author author V. Bally \ and\ author E. L \"o cherbach ,\ @noop journal journal arXiv preprint \ ( year 2026 ) ,\ https://arxiv.org/abs/2606.23433 arXiv:2606.23433 NoStop

    2026 arXiv

  7. [19]

    Bapst , author T

    author author V. Bapst , author T. Keck , author A. Grabska-Barwi \'n ska , author C. Donner , author E. D. \ Cubuk , author S. S. \ Schoenholz , author A. Obika , author A. W. R. \ Nelson , author T. Back , author D. Hassabis ,\ and\ author P. Kohli ,\ https://doi.org/10.1038...

    doi

  8. [21]

    Cang , author J

    author author H. Cang , author J. Li , author V. N. \ Novikov ,\ and\ author M. D. \ Fayer ,\ https://doi.org/10.1063/1.1568338 journal journal The Journal of Chemical Physics \ volume 118 ,\ pages 9303 ( year 2003 a ) NoStop

    2003 doi

  9. [22]

    Cang , author J

    author author H. Cang , author J. Li ,\ and\ author M. D. \ Fayer ,\ https://doi.org/10.1016/S0009-2614(02)01542-7 journal journal Chemical Physics Letters \ volume 366 ,\ pages 82 ( year 2002 ) NoStop

    2002 doi

  10. [23]

    Cang , author J

    author author H. Cang , author J. Li , author H. C. \ Andersen ,\ and\ author M. D. \ Fayer ,\ https://doi.org/10.1063/1.2000235 journal journal The Journal of Chemical Physics \ volume 123 ,\ pages 064508 ( year 2005 ) NoStop

    2005 doi

  11. [24]

    Cang , author J

    author author H. Cang , author J. Li , author V. N. \ Novikov ,\ and\ author M. D. \ Fayer ,\ https://doi.org/10.1063/1.1618215 journal journal The Journal of Chemical Physics \ volume 119 ,\ pages 10421 ( year 2003 b ) NoStop

    2003 doi

  12. [25]

    Mori ,\ https://doi.org/10.1143/PTP.33.423 journal journal Prog

    author author H. Mori ,\ https://doi.org/10.1143/PTP.33.423 journal journal Prog. Theor. Phys. \ volume 33 ,\ pages 423 ( year 1965 ) NoStop

    1965 doi

  13. [26]

    Zwanzig ,\ https://doi.org/10.1063/1.1731409 journal journal J

    author author R. Zwanzig ,\ https://doi.org/10.1063/1.1731409 journal journal J. Chem. Phys. \ volume 33 ,\ pages 1338 ( year 1960 ) NoStop

    1960 doi

  14. [27]

    author author W. G \"o tze ,\ https://doi.org/10.1093/acprof:oso/9780199235346.001.0001 title Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory \ ( publisher Oxford University Press ,\ address Oxford ,\ year 2009 ) NoStop

  15. [28]

    author author L. M. C. \ Janssen ,\ https://doi.org/10.3389/fphy.2018.00097 journal journal Frontiers in Physics \ volume 6 ,\ pages 97 ( year 2018 ) NoStop

    2018

  16. [29]

    van Zon \ and\ author J

    author author R. van Zon \ and\ author J. Schofield ,\ @noop journal journal Phys. Rev. E \ volume 65 ,\ pages 011106 ( year 2002 ) ,\ https://arxiv.org/abs/cond-mat/0108029 arXiv:cond-mat/0108029 NoStop

    2002 arXiv

  17. [31]

    Mayer , author K

    author author P. Mayer , author K. Miyazaki ,\ and\ author D. R. \ Reichman ,\ https://doi.org/10.1103/PhysRevLett.97.095702 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 095702 ( year 2006 ) NoStop

    2006 doi

  18. [32]

    author author L. M. C. \ Janssen , author P. Mayer ,\ and\ author D. R. \ Reichman ,\ https://doi.org/10.1088/1742-5468/2016/05/054049 journal journal J. Stat. Mech. \ ,\ pages 054049 ( year 2016 ) ,\ https://arxiv.org/abs/1510.07636 arXiv:1510.07636 NoStop

    2016 arXiv

  19. [38]

    author author I. V. \ Girsanov ,\ https://doi.org/10.1137/1105027 journal journal Theory Probab. Appl. \ volume 5 ,\ pages 285 ( year 1960 ) NoStop

    1960 doi

  20. [43]

    No \'e \ and\ author F

    author author F. No \'e \ and\ author F. N \"u ske ,\ https://doi.org/10.1137/110858616 journal journal Multiscale Model. Simul. \ volume 11 ,\ pages 635 ( year 2013 ) NoStop

    2013 doi

  21. [44]

    Janssen, Liesbeth M. C. and Mayer, Peter and Reichman, David R. , title =. J. Stat. Mech. , year =. doi:10.1088/1742-5468/2016/05/054049 , eprint =

    2016 doi

  22. [45]

    and Luo, Chengjie and Ciarella, Simone and Janssen, Liesbeth M

    Debets, Vincent E. and Luo, Chengjie and Ciarella, Simone and Janssen, Liesbeth M. C. , title =. Phys. Rev. E , volume =. 2021 , doi =. 2108.06829 , archivePrefix =

    2021 arXiv

  23. [46]

    Maimbourg, Thibaud and Kurchan, Jorge and Zamponi, Francesco , title =. Phys. Rev. Lett. , volume =. 2016 , doi =. 1507.03421 , archivePrefix =

    2016 arXiv

  24. [47]

    Kurchan, Jorge and Maimbourg, Thibaud and Zamponi, Francesco , title =. J. Stat. Mech. , pages =. 2016 , doi =. 1512.02186 , archivePrefix =

    2016 arXiv

  25. [48]

    arXiv preprint , year =

    Deng, Yu and Hani, Zaher and Ma, Xiao , title =. arXiv preprint , year =. 2408.07818 , archivePrefix =

    arXiv

  26. [49]

    Europhys

    Szamel, Grzegorz , title =. Europhys. Lett. , volume =. 2010 , doi =. 1004.3526 , archivePrefix =

    2010 arXiv

  27. [50]

    Franz, Silvio and Jacquin, Hugo and Parisi, Giorgio and Urbani, Pierfrancesco and Zamponi, Francesco , title =. Proc. Natl. Acad. Sci. USA , volume =. 2012 , doi =. 1206.2482 , archivePrefix =

    2012 arXiv

  28. [51]

    arXiv preprint , year =

    Nikolaev, Paul , title =. arXiv preprint , year =. 2407.01217 , archivePrefix =

    arXiv

  29. [52]

    arXiv preprint , year =

    Li, Lei and Wang, Yuelin and Wang, Yuliang , title =. arXiv preprint , year =. 2312.00339 , archivePrefix =

    arXiv

  30. [53]

    Electron

    Lacker, Daniel , title =. Electron. J. Probab. , volume =. 2022 , doi =. 2105.03238 , archivePrefix =

    2022 arXiv

  31. [54]

    Lacker, Daniel , title =. Probab. Math. Phys. , volume =. 2023 , doi =. 2105.02983 , archivePrefix =

    2023 arXiv

  32. [55]

    Propagation of chaos and residual dependence in

    Jalowy, Jonas and Kabluchko, Zakhar and L. Propagation of chaos and residual dependence in. Math. Phys. Anal. Geom. , volume =. 2025 , doi =. 2410.08004 , archivePrefix =

    2025 arXiv

  33. [56]

    Increasing propagation of chaos for mean field models , journal =

    Ben Arous, G. Increasing propagation of chaos for mean field models , journal =. 1999 , doi =

    1999

  34. [57]

    and Newman, Charles M

    Ellis, Richard S. and Newman, Charles M. , title =. Z. Wahrscheinlichkeitstheorie verw. Gebiete , volume =. 1978 , doi =

    1978

  35. [58]

    Bolthausen, Erwin , title =. Z. Wahrscheinlichkeitstheorie verw. Gebiete , volume =. 1986 , doi =

    1986

  36. [59]

    Kusuoka, Shigeo and Liang, Song , title =. Probab. Theory Relat. Fields , volume =. 2000 , doi =

    2000

  37. [60]

    Albeverio, Sergio and Liang, Song , title =. Ann. Probab. , volume =. 2005 , doi =. math/0503601 , archivePrefix =

    2005 arXiv

  38. [61]

    Latz, Arnulf , title =. J. Phys.: Condens. Matter , volume =. 2000 , doi =. cond-mat/9911025 , archivePrefix =

    2000 arXiv

  39. [62]

    Europhys

    Biroli, Giulio and Bouchaud, Jean-Philippe , title =. Europhys. Lett. , volume =. 2004 , doi =. cond-mat/0401260 , archivePrefix =

    2004 arXiv

  40. [63]

    , title =

    Biroli, Giulio and Bouchaud, Jean-Philippe and Miyazaki, Kunimasa and Reichman, David R. , title =. Phys. Rev. Lett. , volume =. 2006 , doi =. cond-mat/0605733 , archivePrefix =

    2006 arXiv

  41. [64]

    , title =

    Berthier, Ludovic and Biroli, Giulio and Bouchaud, Jean-Philippe and Kob, Walter and Miyazaki, Kunimasa and Reichman, David R. , title =. J. Chem. Phys. , volume =. 2007 , doi =. cond-mat/0609656 , archivePrefix =

    2007 arXiv

  42. [65]

    Mori, Hazime , title =. Prog. Theor. Phys. , volume =. 1965 , doi =

    1965

  43. [66]

    Zwanzig, Robert , title =. J. Chem. Phys. , volume =. 1960 , doi =

    1960

  44. [67]

    van Zon, Ramses and Schofield, Jeremy , title =. Phys. Rev. E , volume =. 2002 , eprint =

    2002

  45. [68]

    Luo, Chengjie and Janssen, Liesbeth M. C. , title =. J. Chem. Phys. , volume =. 2020 , doi =. 1909.00428 , archivePrefix =

    2020 arXiv

  46. [69]

    Ikeda, Atsushi and Miyazaki, Kunimasa , title =. Phys. Rev. Lett. , volume =. 2010 , doi =. 1003.5472 , archivePrefix =

    2010 arXiv

  47. [70]

    arXiv preprint , year =

    Zhu, Yuanran , title =. arXiv preprint , year =. 2109.01990 , archivePrefix =

    arXiv

  48. [71]

    Widder, Christoph and Zimmer, Johannes and Schilling, Tanja , title =. J. Phys. A , year =. doi:10.1088/1751-8121/ae02cc , eprint =

    doi

  49. [72]

    arXiv preprint , year =

    Widder, Christoph , title =. arXiv preprint , year =. 2607.25855 , archivePrefix =

    arXiv

  50. [73]

    Emergence of common noise: quantitative conditional propagation of chaos , journal =

    Bally, Vlad and L. Emergence of common noise: quantitative conditional propagation of chaos , journal =. 2026 , eprint =

    2026

  51. [74]

    Laudicina, Corentin C. L. and Janssen, Liesbeth M. C. and Szamel, Grzegorz , title =. Phys. Rev. Lett. , year =. doi:10.1103/8lw2-352q , eprint =

    doi

  52. [75]

    Ikeda, Harukuni and Ikeda, Atsushi , title =. J. Stat. Mech. , pages =. 2016 , doi =. 1603.06314 , archivePrefix =

    2016 arXiv

  53. [76]

    Parisi, Giorgio and Rizzo, Tommaso , title =. Phys. Rev. E , volume =. 2013 , doi =. 1205.3360 , archivePrefix =

    2013 arXiv

  54. [77]

    , title =

    Widmer-Cooper, Asaph and Harrowell, Peter and Fynewever, H. , title =. Phys. Rev. Lett. , volume =. 2004 , doi =

    2004

  55. [78]

    Tong, Hua and Tanaka, Hajime , title =. Phys. Rev. X , volume =. 2018 , doi =

    2018

  56. [79]

    , title =

    Berthier, Ludovic and Jack, Robert L. , title =. Phys. Rev. E , volume =. 2007 , doi =. 0706.1044 , archivePrefix =

    2007 arXiv

  57. [80]

    Unveiling the Predictive Power of Static Structure in Glassy Systems , journal =

    Bapst, Victor and Keck, Thomas and Grabska-Barwi. Unveiling the Predictive Power of Static Structure in Glassy Systems , journal =. 2020 , doi =

    2020

  58. [81]

    Rotation-Equivariant Graph Neural Networks for Learning Glassy Liquids Representations , journal =

    Pezzicoli, Francesco Saverio and Charpiat, Guillaume and Landes, Fran. Rotation-Equivariant Graph Neural Networks for Learning Glassy Liquids Representations , journal =. 2024 , doi =. 2211.03226 , archivePrefix =

    2024 arXiv

  59. [82]

    A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems , journal =

    No. A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems , journal =. 2013 , doi =

    2013

  60. [83]

    Field Theoretic Formulation of a Mode-Coupling Equation for Colloids , journal =

    Jacquin, Hugo and van Wijland, Fr. Field Theoretic Formulation of a Mode-Coupling Equation for Colloids , journal =. 2011 , doi =. 1103.5242 , archivePrefix =

    2011 arXiv

  61. [84]

    Kipnis, Claude and Varadhan, S. R. S. , title =. Commun. Math. Phys. , volume =. 1986 , doi =

    1986

  62. [85]

    Franosch, T. and G. Structure and Structure Relaxation , journal =. 1998 , doi =. cond-mat/9807017 , archivePrefix =

    1998 arXiv

  63. [86]

    and Voigtmann, Th

    Franosch, T. and Voigtmann, Th. , title =. J. Stat. Phys. , volume =. 2002 , doi =. cond-mat/0202238 , archivePrefix =

    2002 arXiv

  64. [87]

    Szamel, Grzegorz , title =. Prog. Theor. Exp. Phys. , volume =. 2013 , doi =. 1209.6622 , archivePrefix =

    2013 arXiv

  65. [88]

    Kurchan, Jorge and Parisi, Giorgio and Urbani, Pierfrancesco and Zamponi, Francesco , title =. J. Phys. Chem. B , volume =. 2013 , doi =. 1303.1028 , archivePrefix =

    2013 arXiv

  66. [89]

    Arita, Chikashi and Krapivsky, P. L. and Mallick, Kirone , title =. Phys. Rev. E , volume =. 2017 , doi =. 1611.07719 , archivePrefix =

    2017 arXiv

  67. [90]

    arXiv preprint , year =

    Dechant, Andreas , title =. arXiv preprint , year =. 2306.00417 , archivePrefix =

    arXiv

  68. [91]

    Wu, Jianlan and Cao, Jianshu , title =. Phys. Rev. Lett. , volume =. 2005 , doi =

    2005

  69. [92]

    Wu, Jianlan and Cao, Jianshu , title =. J. Phys. Chem. B , volume =. 2004 , doi =

    2004

  70. [93]

    Anderson, W. N., Jr. and Duffin, R. J. , title =. J. Math. Anal. Appl. , volume =. 1969 , doi =

    1969

  71. [94]

    Kubo, Fumio and Ando, Tsuyoshi , title =. Math. Ann. , volume =. 1980 , doi =

    1980

  72. [95]

    Iannazzo, Bruno , title =. Numer. Linear Algebra Appl. , volume =. 2016 , doi =

    2016

  73. [96]

    Szamel, Grzegorz , title =. J. Chem. Phys. , volume =. 2007 , doi =. 0705.3645 , archivePrefix =

    2007 arXiv

  74. [97]

    Pihlajamaa, Ilian and Janssen, Liesbeth M. C. , title =. Phys. Rev. Research , volume =. 2024 , doi =. 2408.11579 , archivePrefix =

    2024 arXiv

  75. [98]

    and Laudicina, Corentin C

    Pihlajamaa, Ilian and Debets, Vincent E. and Laudicina, Corentin C. L. and Janssen, Liesbeth M. C. , title =. SciPost Phys. , volume =. 2023 , doi =. 2307.03443 , archivePrefix =

    2023 arXiv

  76. [99]

    Physica A , volume =

    Kawasaki, Kyozi , title =. Physica A , volume =. 1995 , doi =

    1995

  77. [100]

    Classical and Quantum Dynamics in Condensed Phase Simulations , editor =

    Kapral, Raymond and Consta, Styliani and McWhirter, Liam , title =. Classical and Quantum Dynamics in Condensed Phase Simulations , editor =. 1998 , doi =

    1998

  78. [101]

    arXiv:2512.10256 , year =

    Lang, Quanjun and Lu, Jianfeng , title =. arXiv:2512.10256 , year =. 2512.10256 , archivePrefix =

  79. [102]

    Lang, Simon and Schilling, Rolf and Krakoviack, Vincent and Franosch, Thomas , title =. Phys. Rev. E , volume =. 2012 , doi =. 1208.1929 , archivePrefix =

    2012 arXiv

  80. [103]

    and Ciarella, Simone and

    Biezemans, Rutger A. and Ciarella, Simone and. Glassy dynamics from generalized mode-coupling theory: existence and uniqueness of solutions for hierarchically coupled integro-differential equations , journal =. 2020 , doi =. 2006.04476 , archivePrefix =

    2020 arXiv

  81. [104]

    Liu, Wei and Su, Yu and Wang, Yao and Dou, Wenjie , title =. Phys. Rev. Lett. , volume =. 2025 , doi =. 2407.01923 , archivePrefix =

    2025 arXiv

  82. [105]

    Forrester, P. J. and Jancovici, B. , title =. Physica A , volume =. 1997 , doi =

    1997

  83. [106]

    Banchio, Adolfo J. and N. Collective diffusion, self-diffusion and freezing criteria of colloidal suspensions , journal =. 2000 , doi =

    2000

  84. [107]

    Ackerson, B. J. , title =. The Journal of Chemical Physics , volume =. 1976 , doi =

    1976

  85. [108]

    Physical Review E , volume =

    Yeomans-Reyna, Laura and Medina-Noyola, Magdaleno , title =. Physical Review E , volume =. 2000 , doi =

    2000

  86. [109]

    Physica A , volume =

    Cichocki, Bogdan and Hess, Walter , title =. Physica A , volume =. 1987 , doi =

    1987

  87. [110]

    Vaccarino, Francesco , title =. Ann. Inst. Fourier , volume =. 2005 , doi =

    2005

  88. [111]

    , title =

    Mayer, Peter and Miyazaki, Kunimasa and Reichman, David R. , title =. Phys. Rev. Lett. , volume =. 2006 , doi =

    2006

  89. [112]

    Monotonicity, Bounds and Acceleration of Block

    Zimmerling, J. Monotonicity, Bounds and Acceleration of Block. Journal of Scientific Computing , volume =. 2025 , doi =. 2407.21505 , archivePrefix =

    2025 arXiv

  90. [113]

    Shohat, James Alexander and Tamarkin, Jacob David , title =

  91. [114]

    , title =

    Benzi, Michele and Golub, Gene H. , title =. BIT Numerical Mathematics , volume =. 1999 , doi =

    1999

  92. [115]

    BIT Numerical Mathematics , volume =

    Frommer, Andreas and Schweitzer, Marcel , title =. BIT Numerical Mathematics , volume =. 2016 , doi =

    2016

  93. [116]

    Proceedings of the 33rd International Conference on Machine Learning , series =

    Li, Chengtao and Sra, Suvrit and Jegelka, Stefanie , title =. Proceedings of the 33rd International Conference on Machine Learning , series =. 2016 , eprint =

    2016

  94. [117]

    and Stiefel, Eduard , title =

    Hestenes, Magnus R. and Stiefel, Eduard , title =. Journal of Research of the National Bureau of Standards , volume =. 1952 , doi =

    1952

  95. [118]

    BIT Numerical Mathematics , volume =

    Massei, Stefano and Robol, Leonardo , title =. BIT Numerical Mathematics , volume =. 2021 , doi =. 1908.02032 , archivePrefix =

    2021 arXiv

  96. [119]

    Ilhan, Beybin and Mugele, Frieder and Duits, Michael H. G. , title =. Journal of Colloid and Interface Science , volume =. 2022 , doi =. 2012.11729 , archivePrefix =

    2022 arXiv

  97. [120]

    Physical Review Letters , volume =

    Liang, Jian and Feng, Xuan and Zheng, Ning and Wang, Huaguang and Ni, Ran and Zhang, Zexin , title =. Physical Review Letters , volume =. 2025 , doi =. 2407.11479 , archivePrefix =

    2025 arXiv

  98. [121]

    Ranganath , title =

    Nayak, P. Ranganath , title =. Journal of Lubrication Technology , volume =. 1971 , doi =

    1971

  99. [122]

    Effective Boundary Condition at a Rough Surface Starting from a Slip Condition , journal =

    Dalibard, Anne-Laure and G. Effective Boundary Condition at a Rough Surface Starting from a Slip Condition , journal =. 2011 , doi =. 1008.5030 , archivePrefix =

    2011 arXiv

  100. [123]

    Arendt, Wolfgang and ter Elst, A. F. M. , title =. J. Differential Equations , volume =. 2011 , doi =. 1005.0875 , archivePrefix =

    2011 arXiv

  101. [124]

    arXiv preprint , year =

    Li, Liping , title =. arXiv preprint , year =. 2202.10914 , archivePrefix =

    arXiv

  102. [125]

    , title =

    Grebenkov, Denis S. , title =. Phys. Rev. E , volume =. 2019 , doi =

    2019

  103. [126]

    Burdzy, Krzysztof and Chen, Zhen-Qing and Pal, Soumik , title =. Ann. Appl. Probab. , volume =. 2011 , doi =

    2011

  104. [127]

    Ericok, O. B. and Ganesan, K. and Mason, J. K. , title =. Phys. Rev. E , volume =. 2021 , doi =. 2106.10315 , archivePrefix =

    2021 arXiv

  105. [128]

    Acta Math

    Huang, Lu-Jing and Mao, Yong-Hua and Wang, Tao , title =. Acta Math. Sin. Engl. Ser. , volume =. 2023 , doi =. 2106.00324 , archivePrefix =

    2023 arXiv

  106. [129]

    Beta relaxation in supercooled liquids , journal =

    G. Beta relaxation in supercooled liquids , journal =. 1991 , doi =

    1991

  107. [130]

    Physical Review X , volume =

    Rusciano, Francesco and Pastore, Raffaele and Greco, Francesco and Kob, Walter , title =. Physical Review X , volume =. 2026 , doi =

    2026

  108. [131]

    Janssen, Liesbeth M. C. and Mayer, Peter and Reichman, David R. , title =. Journal of Statistical Mechanics: Theory and Experiment , pages =. 2016 , doi =

    2016

  109. [132]

    Angell, C. A. , title =. Science , volume =. 1995 , doi =

    1995

  110. [133]

    and Stillinger, Frank H

    Debenedetti, Pablo G. and Stillinger, Frank H. , title =. Nature , volume =. 2001 , doi =

    2001

  111. [134]

    Reviews of Modern Physics , volume =

    Berthier, Ludovic and Biroli, Giulio , title =. Reviews of Modern Physics , volume =. 2011 , doi =

    2011

  112. [135]

    Pusey, P. N. and van Megen, W. , title =. Nature , volume =. 1986 , doi =

    1986

  113. [136]

    , title =

    Kob, Walter and Andersen, Hans C. , title =. Physical Review E , volume =. 1995 , doi =

    1995

  114. [137]

    and Underwood, S

    van Megen, W. and Underwood, S. M. , title =. Physical Review E , volume =. 1994 , doi =

    1994

  115. [138]

    Relaxation Processes in Supercooled Liquids , journal =

    G. Relaxation Processes in Supercooled Liquids , journal =. 1992 , doi =

    1992

  116. [139]

    , title =

    Anderson, Philip W. , title =. Science , volume =

  117. [140]

    Bengtzelius, U. and G. Dynamics of Supercooled Liquids and the Glass Transition , journal =. 1984 , doi =

    1984

  118. [141]

    , title =

    Leutheusser, E. , title =. Physical Review A , volume =. 1984 , doi =

    1984

  119. [142]

    and Underwood, S

    van Megen, W. and Underwood, S. M. , title =. Physical Review Letters , volume =. 1993 , doi =

    1993

  120. [143]

    Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory , publisher =

    G. Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory , publisher =. 2009 , doi =

    2009

  121. [144]

    Janssen, Liesbeth M. C. , title =. Frontiers in Physics , volume =. 2018 , doi =

    2018

  122. [145]

    , title =

    Ediger, Mark D. , title =. Annual Review of Physical Chemistry , volume =. 2000 , doi =

    2000

  123. [146]

    and Biroli, G

    Berthier, L. and Biroli, G. and Bouchaud, J.-P. and Cipelletti, L. and El Masri, D. and L'H. Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition , journal =. 2005 , doi =

    2005

  124. [147]

    Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability , volume =

    Kac, Mark , title =. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability , volume =

  125. [148]

    , title =

    McKean, Henry P. , title =. Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7) , pages =

  126. [149]

    , title =

    Girsanov, Igor V. , title =. Theory Probab. Appl. , volume =. 1960 , doi =

    1960

  127. [150]

    and Fayer, Michael D

    Cang, Hu and Li, Jie and Andersen, Hans C. and Fayer, Michael D. , title =. The Journal of Chemical Physics , volume =. 2005 , doi =

    2005

  128. [151]

    Cang, Hu and Li, Jie and Novikov, V. N. and Fayer, Michael D. , title =. The Journal of Chemical Physics , volume =. 2003 , doi =

    2003

  129. [152]

    , title =

    Cang, Hu and Li, Jie and Fayer, Michael D. , title =. Chemical Physics Letters , volume =. 2002 , doi =

    2002

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