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On weak ergodicity breaking in mean-field spin glasses
by Giampaolo Folena, Francesco Zamponi
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|Authors (as registered SciPost users):||Giampaolo Folena|
|Preprint Link:||https://arxiv.org/abs/2303.00026v1 (pdf)|
|Date submitted:||2023-03-03 23:30|
|Submitted by:||Folena, Giampaolo|
|Submitted to:||SciPost Physics|
The weak ergodicity breaking hypothesis postulates that out-of-equilibrium glassy systems lose memory of their initial state despite being unable to reach an equilibrium stationary state. It is a milestone of glass physics, and has provided a lot of insight on the physical properties of glass aging. Despite its undoubted usefulness as a guiding principle, its general validity remains a subject of debate. Here, we present evidence that this hypothesis does not hold for a class of mean-field spin glass models. While most of the qualitative physical picture of aging remains unaffected, our results suggest that some important technical aspects should be revisited.
Submission & Refereeing History
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- Report 3 submitted on 2023-04-27 13:14 by Anonymous
- Report 1 submitted on 2023-04-26 12:57 by Anonymous
Reports on this Submission
- Cite as: Anonymous, Report on arXiv:2303.00026v1, delivered 2023-04-27, doi: 10.21468/SciPost.Report.7110
The manuscript address the validity of scenario that has become part of the Spin-Glass folklore in the last thirty years: the idea that off-equilibrium dynamics in mean-field spin-glasses with one step of Parisi's Replica-Symmetry-Breaking explores in a thermal fashion the most numerous marginal states. This idea came from observations on pure p-spin models by Cugliandolo and Kurchan (CK) and is now challenged by recent work including notably this manuscript.
More precisely the authors address the topic of weak versus strong ergodicity breaking, namely if the correlation with initial condition is lost during aging. Based on their results they favor a strong ergodicity breaking scenario while correctly acknowledging that the time regime explored is limited and thus the question cannot be considered settled: "The possibility that the scenario we propose is only a pre-asymptotic regime that would crossover to a weak ergodicity regime thus remains open."
The manuscript discusses a number of features and interesting questions
in a very clear fashion and it is carefully written. It deserves publication
in its present form and I only have a few recommendation.
Quite simply the problem is wether the correlation computed at finite time extrapolate to zero or not at infinite time. If one uses power-laws to extrapolate a finite value is obtained but there is no guarantee that the asymptotic behavior is described by a power-law and not by slower logarithmic decays. Two essential open problems are i) the lack of an analytic solution and ii) the lack of a numerical algorithm capable of reaching large times using a time grid with varying spacing.
Considering a different type of microscopic dynamics, the author show that large times dynamics seems to be independent of the short time details and
this indeed gives hope that an analytic solution can indeed be found and that some of the properties of the CK solution remains valid.
As for the second issue, the authors cite Ref. 60, where one such algorithm was used to reach times of order 10^7, and the reader may be puzzled by why they used a fixed spacing algorithm reaching times of order 10^3, a comment on this seems appropriate.
Ref. 22 is misquoted in the introduction as supporting strong ergodicity breaking but it actually takes an agnostic point of view pointing to the difficulties of extrapolating to large times and urging for an analytical solution. It would be interesting to plot the data of fig. 5 and fig. 6 parametrically and see what would be the asymptotic energy if weak ergodicity breaking was correct, as done in fig. 7 of Ref. 22.
After equation (16) and again in Section G it is mentioned that:" In the case of a quench to the critical temperature, exact relations between the αE and αC exponents were found in Ref. ." It is true that Ref. 54 is at present the only case beyond the p=2 spherical model where the asymptotic behavior of one-time quantities is computed analytically, nonetheless it should be stated that it deals with systems with continuous RSB transitions and not with the discontinuous 1RSB systems studied here.
Also, the comment of Dr. Theo Nieuwenhuizen should be taken into account.
- Cite as: Anonymous, Report on arXiv:2303.00026v1, delivered 2023-04-26, doi: 10.21468/SciPost.Report.7105
The authors analyze mixed spherical random (p+s)-spin glass models (p=2 and 3) undergoing gradient descent dynamics from random initial condition. Numerical integration of the DMFT equations suggests that the weak ergodicity breaking hypothesis does not hold in the mixture models under consideration, at variance with the pure p-spin case. This observation is confirmed by a time series expansion of the overlap with the initial condition C(t,0), reaching a non-zero asymptotic value. A similar expansion of the radial reaction shows that the dynamics approaches a marginally stable minimum, while the asymptotic energy remains higher than the threshold value at which typical minima become marginal.
Overall, these results suggest that strong ergodicity breaking is verified in mixed (p+s)-spin models at any s>p and even from infinite initial temperature, in contradiction with the result of ref.  where strong ergodicity breaking was found only below a finite initial onset temperature. The dynamics appears to find an aging state confined to an initialization-dependent manifold sampled in an effectively thermal way, as shown by the presence of an effective FDR.
This work inscribes in the timely effort to understand and elucidate a long-standing picture of the out-of-equilibrium dynamics of prototypical mean-field models of the glass transition. The previous literature appears to be cited correctly. The paper is of good technical quality and the analytical derivations are well explained and relatively easy to follow. The presentation is generally good, although some aspects could be improved or clarified (see comments below). For these reasons, I recommend publication of the manuscript in SciPost provided the comments below are taken into account.
Comments and questions:
- The analytic derivations are written having an expert reader in mind and frequently referring to other works, sometimes to the detriment of a self-contained presentation. E.g., the “overlap” and the “characteristic polynomial” are introduced on page 3 without definition, and similarly the “complexity” on page 4.
- The coefficients of the mixture seem to play an important role in discriminating between different asymptotic regimes. It would be useful to clarify how sensitive the discrepancy observed, e.g., in Fig. 2 is to the choice of the coefficient \lambda. It would be useful to report the values of the coefficients used in ref.  for comparison.
- It would be interesting if the authors could comment on the implications of their findings for optimization algorithms in planted models, and in relation to the expected discrepancy in performance (if any) between random and “smart” initializations. E.g., should the picture presented in [a,b] be revisited (in particular regarding the (2+4) planted model)? See references below.
- Ref.  supports the finding of a finite onset temperature referring to numerical experiments of realistic glass-forming liquids, where this threshold is observed. How do the authors reconcile this experimental observation with their findings? Does the experimental T_onset show any finite-size dependence? Moreover, according to the phase diagram in Fig. 1(b) of this paper, the 1RSB dynamical ansatz used in ref.  to derive T_onset=0.91 for the (3+4)-spin model does not seem unreasonable, in addition to the good agreement with numerical extrapolation. A clarification on why the asymptotic approximation used in  is wrong and T_onset instead should be set to infinity would be really appreciated.
- The DMFT equations (14) starting from random initialization do not exhibit any explicit dependence on the initial condition via C(t,0). From the presentation in section C, it is not intuitive to me why this dependence should appear in the asymptotic dynamics and I would appreciate a clarification on this point. Moreover, at first one may wonder why the CK ansatz does not apply in this case. I think this point is further clarified in section F, however it could be hard for non-expert readers to connect the dots between sections.
[a] Mannelli, S. S., Biroli, G., Cammarota, C., Krzakala, F., Urbani, P., & Zdeborová, L. (2020). Marvels and pitfalls of the langevin algorithm in noisy high-dimensional inference. Physical Review X, 10(1), 011057.
[b] Sarao Mannelli, S., Biroli, G., Cammarota, C., Krzakala, F., & Zdeborová, L. (2019). Who is afraid of big bad minima? analysis of gradient-flow in spiked matrix-tensor models. Advances in Neural Information Processing Systems, 32.