Dynamical Instantons and Activated Processes in Mean-Field Glass Models

Technical paper of very high level. This is a (very valuable) step towards a very difficult task.

That it does not solve the full problem, but that is very very hard!

The authors build upon results on the free-energy landscape of disordered p-spin models that they derived in recent years, to make a step towards the (very difficult) analytical study of thermally activated processes in models with rugged free-energy landscapes. More precisely, they obtain the dynamical instanton corresponding to a particular process, the passage from a local minimum to another one via a saddle of rank-1 in the spherical p-spin disordered model. This is a very interesting and useful contribution to the theory of the dynamics of complex systems and it merits publication in SciPost. Below I list some changes that I think the authors should implement in the preprint before its publication.

Having chosen a minimum, the saddle and the other minimum, where is the difference with a “normal” instanton calculation? I think the authors should stress in which sense this calculation is different and what makes it special.

The dynamics one is trying to characterise are the ones in a rugged free-energy landscape. In the studies in which the initial states were drawn from the Boltzmann equilibrium pdf, this was ensured. Here, the initial states are chosen with respect to the potential energy landscape. This simplification, that one can accept in the pure p-spin model due to the simplicity of the temperature effects on the free-energy landscape, can diminish the relevance of this calculation for other disordered models. The authors should discuss this important point somewhere in the text. (If they have already done it, I missed it.)

The authors might find useful to compare their results to the simulations in D. A. Stariolo and L. F. Cugliandolo, Phys. Rev. E 102, 022126 (2020) where the MC dynamics of the Ising (instead of spherical) p-spin model was studied and properties of subsequent minima, their overlap, and the barriers crossed, were studied. In particular, the back and forth motion between nearby minima was observed numerically in this paper.

Notation requirements:
It would be convenient to call _m the minima and _s the saddles. Following the subindices is hard, very quickly one gets to the point of asking one-self who was _1, who was _2? Moreover, the choice of _\infty is not happy either, I would say. Instead, Fig. 1 is very useful indeed.
In eq. (29) the integrals over s and s’ should be integrals over times, right? So please use time notation, s was used for something else beforehand. The same in App. D and the following equations.
It’s a pity the overflows in eq. (49) and maybe also in other equations, please check and arrange.

Sec. 2.2
Through the saddles -> through the saddle, right? Because the saddle was also chosen. A clarification of this point is needed.

The fact that the response equation [eq. (55) in the preprint] is ``resilient’’ to changes in the initial conditions, etc. is a quite common fact. The authors could comment on this.

A grammar check would be welcome. I spotted a few English mistakes such as:
- Systems jumps -> systems jump in page 2
one of the main aim -> one of the main aims in page 6
not clear what is the need for the phrase “- an information that is missing so far” also in page 6.
Is ``to transition’’ a verb? Page 19.
Surely there are others.

See the report.

See attached report

Essentially none

I fully recommend publication in its present form

This article further deepens the knowledge on activated events in complex mean-field energy landscapes, giving new mathematical tools that may open the path to new discoveries in the field. The focus is on the pure (homogeneous) p-spin spherical, which in the last years has received an increasing interest within both the mathematical and physical community.
The main result is the possibility of geometrically connecting two minima of the energy landscape through a saddle and deriving the mean-field “dynamical instantons” equations that control this barrier jumping.
In my opinion, the greatest achievement consists in evaluating the dynamical mean-field equations for correlation and response, given some highly-non-trivial conditioning, i.e. initiating the system in a given saddle at a given distance from a reference minimum.

The Abstract is clear and exhaustive, however, I would mention that the analyzed model is the pure (homogeneous) p-spin. The fact that the pure model is considered, allows to rewrite the mean-field action in terms of matrix elements and therefore allows the simplification used when conditioning (Eq. 18 and 19). Moreover, all the general comments about the energy landscape and the presence of an energy threshold are valid concepts only in pure models.

1. The introduction is very well written.
Nevertheless, some small comments are needed.
The phrase “mean-field models displays two dynamical regimes: [...]approaches ( or more precisely ages toward) the threshold states. [...]” is misleading since in general there is no unique threshold. Again the homogeneous p-spin models are not generic mean-field models.
Moreover, when introducing the dynamical instanton, I would be more careful to precisely introduce the idea, since, for what I have grasped, it is a new concept; and in the phrase “This is different with respect to standards phase transition […]”, maybe instead of ‘standard’ the word ‘equilibrium’ can be used? It is the concept of equilibrium and out-of-equilibrium connected to the definition of dynamical instanton?

2. The Summary of results
here all the main results are presented in a very simple form, without technicalities.
In fig.3 I would suggest some isoclines to better see the variations.
Still, I am a bit confused by the dynamical instanton definition. Shouldn’t the correlation c(t,t’) plotted in fig.4 depend on the dimension N of the system? Does the temperature enter in the definition of dynamical instanton?

3. Self-consistent …
this is the core of the article where the main analytical results are derived. Here the mean-field dynamical equations for correlation and response of the dynamics conditioned to start from a saddle at a given distance from a reference minimum are derived. The calculations presented are not self-contained and some results were derived by the same authors in previous articles. The choice on how to split the calculations between section 3 and the appendices is a bit arbitrary, and in my opinion, the reader cannot easily follow the steps going back and forth. I would suggest to directly move the formulas presented in subsection 3.1.3 in the appendices and just give a comment on the term K(s,s^) in the main text.
In page 14 “A standard calculation gives:” would be nice to have a reference.
In equation 27 an integral is missing.
In equation 29, S(2) → S(1). And maybe would be better to use t,t’ for times variables
Section 3.3 is clear.
I think that this is the most important part of this work and I would have taken more care in the explanation of the calculations in the hope that this could be used as a starting point for many other similar calculations by other researchers in the field. On the other hand, I understand the technical difficulties to explain in details such a calculation. Therefore, this is only a suggestion.

4. Where does the system fall…
this section presents the asymptotic analysis of the conditioned mean-field equations making use of the TTI and FDT assumptions (stationary limit), allowing to connect the reference saddle with the two minima on the two side of the barriers. This is a standard calculation and the results are very well exposed.
I suggest to write down the energy of the first minimum in all the figures.
I appreciated fig. 7 (right). Perfect.

5. Numerical solution …
this section confirms the asymptotic results obtained in section 4. The kick procedure is clear at the analytical level, however, I had some difficulty following the numerical implementation of the kick.
In subsection 5.2 it is not true that to evaluate c(t,t’) one needs only c(t-dt,s) for all s<t-dt, but rather every point c(s,s’) with s<t’ and s<t. The subsequent discussion on the kick implementation is not quite clear.
Also, I would cite some previous work on the numerical implementation of the integration scheme.
The last phrase in parenthesis “(only the latter has a non-trivial TTI dynamics since a=0)”, what does it mean?

6. The shape of the dynamical instanton ...
despite the fact that this section defines the title of the article, to me it seems less important than the previous ones. The dynamical instanton is evaluated by reversing the downhill dynamics from the saddle and joining it with the dynamics from the saddle to the second minimum. The final result is not the exact instanton but rather an approximation, which merges the two-time sectors.
In Eq. 102 and 103 \tau_S → \tau_s
One point that I think is not clear is how in Fig. 4 the \tau_s, that defines the time of saddle crossing, is chosen. Is \tau_s chosen arbitrarily? and if that is the case, what is the relative dimension N and temperature at which this instantonic solution would be expected to give a contribution to the equilibrium dynamics in a finite-size system?

7. Conclusion
It is maybe the less impressive part of this work. It gives a good resumption of the article but does not contribute to a concrete perspective that these calculations will permit. In particular, it seems that all the efforts in this work bring only a qualitatively understanding: “whereas the ones analysed in this work are more likely to give rise to back and forth motions with frequent returns to the original minimum.” How can the dynamical instanton give some insights into the equilibrium dynamics of a finite size model?

My general opinion is that this article makes fundamental steps forward in the field of mean-field dynamical equations, and activated events in complex landscapes. The analysis is mainly devoted to mathematical aspects rather than the physical ones. The results are very powerful methods to deal with conditioned dynamics; in my opinion, it only lacks the physical perspective a bit. I have appreciated the effort on introducing the subject in the first two sections with very clear and technicality-free literature. I would suggest some more care in the introduction of dynamical instanton and the physical motivations behind its calculation.