Some more small fixes.

1 files changed, 7 insertions, 3 deletions

diff --git a/when_annealed.tex b/when_annealed.tex
index 0b1e52a..e89053c 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -153,7 +153,7 @@ $s>8$ have non-convex $\chi$ and those with $s\leq8$ have convex $\chi$ independ
 of $\lambda$. Second, the characterization of the ground state has been made
 \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
 Crisanti_2011_Statistical, Auffinger_2022_The}. In the $3+s$ models we
-consider, for $s>12.430...$ nontrivial ground state configurations appear in
+consider, for $s>12.430...$ nontrivial ground state configurations (more than {\oldstylenums1\textsc{rsb}}) appear in
 a range of $\lambda$. These bounds on equilibrium order are shown in
 Fig.~\ref{fig:phases}, along with our result for where the complexity has
 nontrivial correlations between some stationary points. As evidenced in that
@@ -206,7 +206,7 @@ complexity has been computed for a couple examples which have nontrivial ground
 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
+negative one means it is vanishingly likely they will appear. The line of
 zero complexity is significant as the transition between many stationary points
 and none.
 
@@ -406,7 +406,11 @@ the complexity that begins at $E_{\oldstylenums1\textsc{rsb}}^+$, the higher
 energy point, ends exactly at $E_{\oldstylenums1\textsc{rsb}}^-$, the lower
 energy point, so that these two points give the precise range of energies at
 which \textsc{rsb} saddles are found. An example that conforms with this
-picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}.
+picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}. In
+that figure, the range of $\mu$ with {\oldstylenums1\textsc{rsb}} ordering at
+any fixed $E$ is extremely small. With increasing $s$ the range also increases
+(see the example of the $3+16$ model in \cite{Kent-Dobias_2023_How}), but we do
+not have any intuition for why this is.
 
 The discriminant $\Delta_f$ inside the square root of \eqref{eq:energies} is
 proportional to