Change addressing report #1, second comment/typo

Added a footnote discussing the commutation of the limits N→∞ and α→0 or
M=1.

2 files changed, 3 insertions, 15 deletions

diff --git a/referee_response.md b/referee_response.md
index 78a3276..258b489 100644
--- a/referee_response.md
+++ b/referee_response.md
@@ -1,21 +1,7 @@
 # Report #1
 
-In this article, the author considers the statistics (mostly the average) of the Euler characteristics for solutions of M
-random constraint satisfaction problems Vk(x)=V0 where the vector x∈√NSN−1 is on the N-sphere. This quantity provides interesting information on the topology of the solution space. The author considers specifically the limit N,M→∞ with fixed ratio α=M/N
-
-. From the expression of the average Euler characteristics, the author obtains a phase diagram for the model, separating various topologies for the space of solutions. The results on the average are further confirmed and extended by computations of the second moment as well as the average logarithm of the Euler characteristics.
-
-As a byproduct of the computations, the author derives interesting conjectures on spherical spin-glass model by taking the limit α,V0→0
-, with E=α−1/2V0 fixed. For M=1
-
-, the solutions of the constraint satisfaction problem indeed coincide with level-sets of the spherical spin-glass.
-
-The article is very well written and reports new and interesting results. I therefore recommend it for publication in SciPost Physics.
-
 Please find below a short list of comments/ typos:
 
--p 10: "This fact mirrors another another that was made clear recently"
-
 -Some of the results derived for the spherical spin-glass stem from the limit mentioned above. It is clear that this limit may be relevant as some important energy scales are recovered that way. Note however that starting from a regime α=M/N=O(1)
 and taking the limit α→0 should in fact match with the limit M→∞ of the regime M=O(1). Thus there is no guarantee that it should provide relevant information on the case M=1. A direct study of the level set of the spherical spin-glass model as done in Appendix D should allow to confirm this prediction. It was not entirely clear from the manuscript whether this analysis indeed does confirm the relevance of Esh
 
diff --git a/topology.tex b/topology.tex
index 281831c..11f1277 100644
--- a/topology.tex
+++ b/topology.tex
@@ -634,7 +634,9 @@ into the structure of solutions in this regime is merited.
 When $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=V_0/\sqrt N$. All the
 results from the previous sections follow, and can be translated to the spin
-glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed. With a little algebra this procedure yields
+glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed.\footnote{
+  It is plausible that the limits of $N\to\infty$ implicit in the saddle point expansion and the limit of $\alpha\to0$ taken here do not commute, and that $M=1$ should be taken from the beginning of the calculation. However, in this case the two procedures do commute. The $\alpha\to0$ limit accomplishes only the elimination of the first term from the effective action \eqref{eq:S.m}, while following Appendix~\ref{sec:euler} with $M=1$ from the outset results in the same term not appearing in the effective action because it is of subleading order in $N$.
+} With a little algebra this procedure yields
 \begin{align}
   E_\text{on}=\pm\sqrt{2f(1)}
   &&