Change addressing report #2, second question

Added a footnote discussing the comparison between our prediction of
V_SAT and that made in the confluent tissues paper.

2 files changed, 4 insertions, 2 deletions

diff --git a/referee_response.md b/referee_response.md
index f2bd481..7f939eb 100644
--- a/referee_response.md
+++ b/referee_response.md
@@ -31,7 +31,7 @@ Publish (meets expectations and criteria for this Journal)
 (ii) It is mentioned that the naive satisfiability threshold predicted from the vanishing of (12) coincides with the threshold obtained within the replica symmetric analysis of the cost function (3). By reading the manuscript, I have missed if/how the satisfiability threshold arising from the analysis of the average Euler characteristics compares with the threshold obtained from the zero-temperature analysis of the equilibrium problem with energy (3): could the Author comment on this?
 
  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.
+ 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.
  3.
  4. Maybe??
 
diff --git a/topology.tex b/topology.tex
index 99c9a06..d3d90d4 100644
--- a/topology.tex
+++ b/topology.tex
@@ -409,7 +409,9 @@ $V_0=V_\text{\textsc{sat}}$ corresponding to the vanishing of the effective
 action at the $m=0$ solution, with $\mathcal S(0)=0$. For a generic covariance
 function $f$ it is not possible to write an explicit formula for
 $V_\text{\textsc{sat}}$, and we calculate it through a numeric
-root-finding algorithm.
+root-finding algorithm.\footnote{
+As a check of this calculation, the satisfiability threshold calculated here can be compared with that calculated using the zero-temperature limit of an equilibrium treatment of the cost function \eqref{eq:cost} made in Ref.~\cite{Urbani_2023_A} for the case where $f(q)=\frac12q^2$ and $\alpha=\frac14$. The authors estimate $V_\text{\textsc{sat}}\simeq1.871$, whereas this manuscript predicts $V_\text{\textsc{sat}}=1.867229\dots$, a seeming inconsistency. However, the author of Ref.~\cite{Urbani_2023_A} indicated in private correspondence that this difference could easily be explained by inaccuracy in the numeric \textsc{pde} treatment of the \textsc{frsb} equilibrium problem. Therefore, this manuscript is consistent with the previous work, but the agreement is not precise.
+}
 
 When $V_0^2<V_\text{sh}^2$, the solution at $m=0$ is difficult to interpret, since
 the action takes a complex value. Such a result could arise from the breakdown