From 4a9455031004be737f10a1a1a786a3de79c623fa Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 22 Jun 2023 11:07:07 +0200
Subject: Final tweaks.

---
 when_annealed.tex | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index fb03c3e..c24f2c4 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -138,7 +138,7 @@ temperature \cite{Crisanti_1992_The}.\footnote{
   between states.
 }
 This is a strong condition on the form of equilibrium order. Note that
-non-convex $\chi$ does not imply that you will see nontrivial correlations between
+non-convex $\chi$ does not imply that you \emph{will} see nontrivial correlations between
 states at some temperature. In the $3+s$ models we consider here, models with
 $s>8$ have non-convex $\chi$ and those with $s\leq8$ have convex $\chi$ independent
 of $\lambda$. Second, the characterization of the ground state has been made
@@ -148,7 +148,7 @@ consider, for $s>12.430...$ nontrivial ground state configurations 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
-figure, correlations among saddles are possible well inside regions which
+figure, correlations among saddles are possible well inside regions that
 forbid them among equilibrium states.
 
 There are two important features which differentiate stationary points
@@ -160,7 +160,7 @@ stability governs the spectrum of the stationary point.
 In each spherical model, the spectrum of every stationary point is a Wigner
 semicircle of the same width $\mu_\mathrm m=\sqrt{4f''(1)}$, but shifted by
 constant. The stability $\mu$ sets this constant shift. When $\mu<\mu_\mathrm
-m$, the spectrum still has support over zero and we have saddles with an
+m$, the spectrum has support over zero and we have saddles with an
 extensive number of downward directions. When $\mu>\mu_\mathrm m$ the spectrum
 has support only over positive eigenvalues, and we have stable minima.\footnote{
   Saddle points with a subextensive number of downward directions also exist
@@ -305,9 +305,10 @@ elsewhere) the constants
   &&
   z_f=f(f''-f')+f'^2
 \end{align}
+When $f$ and its derivatives appear without an argument, the implied argument is always 1, so, e.g., $f'\equiv f'(1)$.
 If $f$ has at least two nonzero coefficients at second order or higher, all of
 these constants are positive. Though in figures we focus on the lower branch of
-saddles, another set of identical solutions always exists for $E\mapsto-E$ and $\mu\mapsto-\mu$.
+saddles, another set of identical solutions always exists for $(E,\mu)\mapsto(-E,-\mu)$.
 We also define $E_\textrm{min}$, the minimum energy at which saddle points with
 an extensive number of downward directions are found, as the energy for which
 $\mu_0(E_\mathrm{min})=\mu_\mathrm m$.
@@ -361,7 +362,7 @@ $e$, $g$, and $h$ are given by
     and a {\oldstylenums1}\textsc{rsb} complexity in necessary. The red points
     show where $\det M=0$. The left point, which is only an upper bound on the
     transition, coincides with it in this case. The gray shaded region
-    highlights the minima, which are stationary points with $\mu>\mu_\mathrm
+    highlights the minima, which are stationary points with $\mu\geq\mu_\mathrm
     m$. $E_\textrm{min}$ is marked on the plot as the lowest energy at which
     extensive saddles are found.
   } \label{fig:complexity_35}
@@ -541,9 +542,9 @@ spherical models has different quenched and annealed averages, as the result of
 nontrivial correlations between stationary points. We saw that these conditions
 can arise among certain populations of saddle points even when the model is
 guaranteed to lack such correlations between equilibrium states, and exist for
-saddle points at a wide range of energies. This suggests that studies using
+saddle points at a wide range of energies. This suggests that studies making
 complexity calculations cannot reliably use equilibrium behavior to defend
-making the annealed approximation. Our result has direct implications for the
+the annealed approximation. Our result has direct implications for the
 geometry of these landscapes, and perhaps could be influential to certain
 out-of-equilibrium dynamics.
 
-- 
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