From f99e3de4414245dd71ddaa492691f7f5c69d6990 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 23 Aug 2023 23:26:24 +0200
Subject: More fixes for referees.

---
 when_annealed.tex | 16 +++++++++-------
 1 file changed, 9 insertions(+), 7 deletions(-)

diff --git a/when_annealed.tex b/when_annealed.tex
index 5e02233..d0323de 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -207,7 +207,7 @@ logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$.
 The annealed complexity has been computed for these models
 \cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched
 complexity has been computed for a couple examples which have nontrivial ground
-states \cite{Kent-Dobias_2023_How}. The annealed complexity bounds the
+states \cite{Crisanti_2006_Spherical ,Kent-Dobias_2023_How}. The annealed complexity bounds the
 complexity from above. A positive complexity indicates the presence of an
 exponentially large number of stationary points of the indicated kind, while a
 negative one means it is vanishingly unlikely they will appear. The line of
@@ -265,7 +265,7 @@ for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb
   \end{aligned}
 \end{equation}
 where $\Delta x=1-x$ and
-\begin{equation}
+\begin{equation} \label{eq:hess.term}
   \mathcal D(\mu)
   =\begin{cases}
     \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
@@ -303,7 +303,7 @@ a(E,\mu)=0$. Going along this line in the replica symmetric solution, the
 $x=q_1=1$ \cite{Kent-Dobias_2023_How}. Since all the parameters in the
 bifurcating solution are known at this point, we can search for it by looking
 for a flat direction. In the annealed solution for
-points describing saddles ($\mu<\mu_\mathrm m$), this line is
+points describing saddles (with $\mu^2\leq\mu_\mathrm m^2$ and therefore the simpler form of \eqref{eq:hess.term}), this line is
 \begin{equation} \label{eq:extremal.line}
   \mu_0=-\frac{2Ef'f''}{z_f}-\sqrt{\frac{2f''u_f}{z_f^2}\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}
 \end{equation}
@@ -472,11 +472,13 @@ smoothly connects $E_{\oldstylenums1\textsc{rsb}}^+$ and
 $E_{\oldstylenums1\textsc{rsb}}^-$, so $E_{\oldstylenums1\textsc{rsb}}^-$
 corresponds to the lower bound of \textsc{rsb} complexity. For large enough
 $s$, the range passes into minima, which is expected as these models have
-nontrivial complexity of their ground states. This also seems to correspond
-with the decoupling of the \textsc{rsb} solutions connected to
+nontrivial complexity of their ground states. Interestingly, this appears to
+happen at precisely the value of $s$ for which nontrivial ground state
+configurations appear, $s=12.403\ldots$. This also seems to correspond with the
+decoupling of the \textsc{rsb} solutions connected to
 $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, with
-the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}. In
-these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower
+the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}.
+In these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower
 bound, but sometimes it is given by the termination of the phase boundary
 extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
 
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