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.

1 files changed, 3 insertions, 1 deletions

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