Cleaned up and reorganized referee response

1 files changed, 1 insertions, 47 deletions

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@@ -6,20 +6,7 @@
 
 # Report #2
 
-The manuscript is clear and self-contained. Below are some questions and requests for clarifications:
-
-(iv) If understand correctly, the vector x0 is arbitrary and it is introduced with the purpose of decomposing the contributions to the Euler characteristics in terms of m. Given the arbitrarily of x0, one would naively expect that the “observable” part of the solution space corresponds to m=0, and that any analysis of the constraint satisfaction problem that is x0-independent should be unable to pick up the transition between Regime II and Regime III: is this the case?
-Requested changes
-
-1- I would clarify the connection to previous results, if available, particularly in relation to points (ii) and (iii) mentioned above.
-
-2- Possibly add some comments on the other points in the report.
-
-3- Consider adding more information to Fig. 2, such as the notation for the values of the order parameters where transitions occur (V_on, V_sh, V_SAT), and the key features of the average Euler characteristics in each regime.
-
-4- Include a comment stating that the functions in equation (7) are Morse, which justifies the use of equation (6). The Smale condition is mentioned, but not explained.
-Recommendation
-
+## Questions and requests for clarifications
 
  1. In fact, the action is not complex when evaluated at m_* for V² > V_on² even though m_* itself becomes complex: the action remains real but becomes negative in this regime. This means that the contribution of these complex-m_* solutions in this regime shrinks with increasing N, and rather than representing a subleading but exponentially large (or even order 1) contribution to the Euler characteristic, their contribution is negligible.
  2. The reference "A continuous constraint satisfaction problem for the rigidity transition in confluent tissues", which performs the FRSB treatment of the zero-temperature equilibrium problem for the case where f(q) = ½ q² and α = ¼, estimates V_SAT ≃ 1.871. Our calculation instead predicts V_SAT = 1.867229…. In private correspondence with the author of the quoted reference, they indicated that such a discrepancy could easily be due to inaccuracy in the numeric PDE treatment of the FRSB equilibrium problem and that they were not concerned by the seeming inconsistency. So, for the moment the two treatments are consistent but the agreement is not precise. A small discussion of this has been added in a footnote to the manuscript.
@@ -36,39 +23,6 @@ Recommendation
 
 # Report #3
 
-
-1- The paper provides insufficient discussion of previous work. The author condenses key background and literature into two brief sentences (at the start of the second paragraph of Section 1 and at the end of page 2). Even for readers familiar with these references, this is hard to parse. I had to go into the bibliography and see which paper the author was referring to in order to follow. I strongly recommend expanding the introduction and discussing prior work in greater detail to provide adequate context for the reader.
-2- The interpretation of magnetization m is unclear. While briefly mentioned at the beginning of Section 2.2, the explanation is insufficient. Since there is no planting in this problem, the physical meaning of an arbitrary random direction is still unclear to me.
-3- Although the introduction to Euler characteristics in Section 2.1 is generally well-presented, I am still uncertain about some aspects:
-3.1- At the beginning of Section 2.2, the author notes compatibility with an N−M−1 sphere, yet the Euler characteristic should be 2 for any hypersphere regardless of dimension. Can something be concluded about dimensionality here?
-3.2- The large Euler characteristic could result from either many disconnected components or the manifold being a product of many manifolds, but the analysis does not distinguish between these cases. How might these scenarios lead to different landscapes? Can we say something about the possible implications for the dynamics?
-4- The connection with the dynamics is not fully convincing. In particular, the theory provided in the paper does not explain the relationship between dynamics and the temperature dependence observed in references [26, 27]. These references identify different behaviours based on the initial temperature in a mixed p-spin model, yet this aspect does not seem to emerge.
-4.1- Additionally, I wonder if the authors have considered how planting would affect the landscape. In the mixed p-spin case, planting simplifies the picture compared to what was observed in [26, 27].
-5- The derivation lacks sufficient detail in some sections. The author uses properties of the superdeterminant without providing references, making it difficult to follow. For example, the steps leading to equations (37-39) are unclear.
-5.1- In equation (47), a superdeterminant with a suffix is introduced without a definition, which makes it challenging to interpret.
-Report
-
-This paper addresses a technically challenging and important problem, making a significant contribution to the study of loss landscapes in constraint satisfaction problems. The innovative application of the Kac-Rice formula and the identification of new regimes add meaningful insights to the field. Although there are some areas requiring clarification and expansion, especially regarding background context, the interpretation of the order parameter, and the connection to dynamics, these revisions mainly pertain to the clarity and depth of exposition rather than fundamental issues. Overall, I recommend acceptance, contingent on addressing the previously mentioned issues in a revised version.
-Requested changes
-
-1- Expand the introduction, providing more context for the problem and discussing relevant previous contributions.
-2- Clarify the physical interpretation of the order parameter m.
-3- Discuss the different scenarios that could result from a large Euler characteristic and their implications for the landscape and dynamics.
-4- Provide further clarity on the connection with dynamics, specifically addressing the temperature dependence observed in prior work and the potential effect of planting.
-5- Provide additional detail in the derivation, particularly in Sections A and B.1.
-Recommendation
-
-Ask for minor revision
-
-    validity: high
-    significance: good
-    originality: high
-    clarity: low
-    formatting: perfect
-    grammar: perfect
-
-2- The interpretation of magnetization m is unclear. While briefly mentioned at the beginning of Section 2.2, the explanation is insufficient. Since there is no planting in this problem, the physical meaning of an arbitrary random direction is still unclear to me.
-
  1. A discussion of the previous literature on this model has been added in the introduction.
  2. A discussion of how to interpret the order parameter *m* has been added to the end of section 2.1.
  3. See the comments below.