On weak ergodicity breaking in mean-field spin glasses

We have followed the suggestion of reviewer 1 and added an appendix which shows the impact of the time step dt .

Referee

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.

Response We agree with the referee that it is of fundamental importance to show that the chosen dt=0.05 gives a small error with respect to the exact dt -> 0 limit. Indeed this is the case, and we substantiated it in an additional appendix A, which provides explicit plots (Fig.13) of the error as a function of the time, showing that the error for dt=0.05 is small and decreases with time, and therefore it does not corrupt the analysis. We remark that that an analogous analysis of the error was provided in a previous article by one of the authors.

Referee

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.

Response As suggested by the referee we have added a new plot of the energy vs the Correlation (Fig.8) and commented accordingly. We remark that the analysis of the error in dt provide evidence that the results shown in Fig.8 should have a 'very small' difference with the exact results in the dt->0 limit. For further confirmation we have added the extroplated (dt->0) line in Fig.8 for the (3+11)-spin model (dashed dotted red line) for comparison.