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On weak ergodicity breaking in mean-field spin glasses
by Giampaolo Folena, Francesco Zamponi
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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.
Author comments upon resubmission
The two reports are positive and only a few small changes and precisations have been requested.
We have carefully considered all recommendations and we have modified the text accordingly.
We think that our manuscript is ready for publication.
List of changes
- properly introduced the “overlap”, the “characteristic polynomial” and the “complexity”.
- added footnote to explain that $\lambda$ is chosen to maximize the “discrepacy”
- added comment on T_onset crossover versus transition.
- added a footnote to comment about previous long time numerical solutions
- rephrased the sentence to answer to the misquotation of 
- clarified that in  the considered transition is of continuous type
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
I thank the authors for their response. I agree with reviewer 1 that a plot showing the impact of the time step dt on the extrapolation is important to support their finding. I believe that with this modification the paper is ready for publication.
- Cite as: Anonymous, Report on arXiv:2303.00026v2, delivered 2023-05-24, doi: 10.21468/SciPost.Report.7242
As I said in the previous report the paper deserves publication. However I ask the authors to make a small additional effort to address an important technical issue. The key ingredient of the paper is the numerical solution of the off-equilibrium dynamical equations by time discretization. Therefore the outcome depends on the time spacing dt. The correct procedure is then to extrapolate the solution to dt=0 considering different values of dt's. It seems that instead the authors report only data at fixed dt=.05 and do not mention the issue at all.
Take for instance fig. 13 (a), it suggests that the numerical curve with dt=.05 does not have large corrections at least up to time t=100 due to the agreement with the results from the series expansion, BUT this is not enough to argue that the agreement will continue to larger times. In order to do so one should compare the result with smaller dt to be sure that the results can be trusted.
Now this is not a minor issue, take for instance the problem I raised in the previous report to which the authors replied:
"We have tried the suggested procedure, i.e. plotting the excess energy versus C(t,0) and performing a 3-parameters fit (see attached figure CorrEn.pdf: (a) for 2-spin and (b) for 3-spin). However, the results remain more consistent with the strong ergodicity breaking scenario, without adding further insights. We thus prefer to not add them to the paper, in order to avoid overcharging it."
Let me note that the interest of such a plot is not to argue if favor or strong ergodicity breaking or not.
Indeed any claim on strong ergodicity breaking is based on extrapolations to infinite time and therefore prone to endless discussions, given that there is no analytical prediction on how the correlation with the initial condition should decay to zero. The aim of the parametric plot is essentially different, the question is: **assuming** that there is weak ergodicity breaking, does the asymptotic energy go to the threshold one? Looking from the figure attached to the reply one would say that this is not the case and this is a very interesting statement.
Therefore I urge the authors to put the figure in the manuscript (there are already 16 figures in the paper, I do not think that one more will change much in terms of readability).
However this is clearly an instance in which one would like to be sure that the data do not suffer from a systematic discretization error. Therefore I would like to see the same parametric curves with a different dt so that the data plotted can be taken confidently to describe the continuum dt=0 limit. The reader must know if the data they are seeing are reliable and this could be done in an appendix.
1) put the correlation-energy parametric figure in the text
2) display the effect of the dicretization dt on the data of the aforementioned figure in a separate figure, to be put in the appendix