Modified article for two columns.

1 files changed, 62 insertions, 49 deletions

diff --git a/when_annealed.tex b/when_annealed.tex
index 282ebc4..37ee7fb 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -169,7 +169,7 @@ has support only over positive eigenvalues, and we have stable minima.\footnote{
 
 \begin{figure}
   \centering
-  \includegraphics{figs/phases_34.pdf}
+  \includegraphics[width=0.95\columnwidth]{figs/phases_34.pdf}
   \caption{
     A phase diagram of the boundaries we discuss in this paper for the $3+s$
     model with $f=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$. The blue region
@@ -223,12 +223,15 @@ tool set, the problem is reduced to the evaluation of an integral by the saddle
 point method for large $N$ \cite{Kent-Dobias_2023_How}. The complexity is given
 by extremizing an effective action,
 \begin{equation}
-  \Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu)
-  =\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)}
-  =\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
+  \begin{aligned}
+    &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \\
+    &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \\
+    &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
+  \end{aligned}
 \end{equation}
-for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by
-\begin{equation}
+for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb.action}.
+\begin{widetext}
+  \begin{equation} \label{eq:1rsb.action}
   \begin{aligned}
     &\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
     =\mathcal D(\mu)
@@ -261,6 +264,7 @@ where $\Delta x=1-x$ and
     -\log\left(\left|\frac{\mu}{\mu_\text m}\right|-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu^2>\mu_\text m^2
   \end{cases}
 \end{equation}
+\end{widetext}
 The details of the derivation of these expressions can be found in \cite{Kent-Dobias_2023_How}.
 The extremal problem in $\hat\beta$, $r_\mathrm d$, $r_1$, $d_\mathrm d$, and
 $d_1$ has a unique solution and can be found explicitly, but the resulting
@@ -298,12 +302,11 @@ elsewhere) the constants
 \begin{align}
   u_f&=f(f'+f'')-f'^2
   &&
-  v_f=f'(f''+f''')-f''^2 \\
+  v_f=f'(f''+f''')-f''^2 \notag \\
   w_f&=2f''(f''-f')+f'f'''
   &&
-  y_f=f'(f'-f)+f''f
-  &&
-  z_f=f(f''-f')+f'^2
+  y_f=f'(f'-f)+f''f \\
+  z_f&=f(f''-f')+f'^2 \notag
 \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
@@ -328,13 +331,15 @@ where $y=-\frac12z_f\mu-f'f''E$ is proportional to the square-root term in
 \eqref{eq:extremal.line} and the constants $a$, $b$, $c$, and $d$ are defined
 by
 \begin{equation}
-  a=\frac{w_f\big(3y_f^2-4ff'f''(f'-f)\big)-6y_f^2(f''-f')f''}{(u_fz_ff'')^2f'}
-  \qquad
-  b=\frac{f'w_f}{z_f^2}
-  \qquad
-  c=\frac{w_f}{f''z_f^2}
-  \qquad
-  d=\frac{w_f}{f'f''}
+  \begin{aligned}
+    a&=\frac{w_f\big(3y_f^2-4ff'f''(f'-f)\big)-6y_f^2(f''-f')f''}{(u_fz_ff'')^2f'}
+    \\
+    b&=\frac{f'w_f}{z_f^2}
+    \qquad
+    c=\frac{w_f}{f''z_f^2}
+    \qquad
+    d=\frac{w_f}{f'f''}
+  \end{aligned}
 \end{equation}
 Changing variables from $\mu$ to $y$ is convenient because the branch
 of \eqref{eq:extremal.line} is chosen by the sign of $y$ (the lower-energy
@@ -351,7 +356,7 @@ $e$, $g$, and $h$ are given by
 
 \begin{figure}
   \centering
-  \includegraphics{figs/complexity_35.pdf}
+  \includegraphics[width=0.95\columnwidth]{figs/complexity_35.pdf}
 
   \caption{
     Stationary point statistics as a function of energy density $E$ and
@@ -372,11 +377,15 @@ The solutions for $\det M=0$ can be calculated explicitly and correspond to
 energies that satisfy
 \begin{equation} \label{eq:energies}
   E_{\oldstylenums1\textsc{rsb}}^\pm
-  =\operatorname{sign}(bg-de)\frac{-cg\pm\sqrt{c^2g^2+(2dh-ag)(bg-de)}}
+  =\frac{\operatorname{sign}(bg-de)\big(-cg\pm\sqrt{\Delta_f}\big)}
   {
-    \sqrt{2c^2eg+(2bh-ae)(bg-de)\mp2ce\sqrt{c^2g^2+(2dh-ag)(bg-de)}}
+    \sqrt{2c^2eg+(2bh-ae)(bg-de)\mp2ce\sqrt{\Delta_f}}
   }
 \end{equation}
+where the discriminant $\Delta_f$ is given by
+\begin{equation}
+  \Delta_f=c^2g^2+(2dh-ag)(bg-de)
+\end{equation}
 This predicts two points where a {\oldstylenums1}\textsc{rsb} solution can
 bifurcate from the annealed one. The remainder of the transition line can be
 found by solving the extremal problem for the action very close to one
@@ -388,16 +397,18 @@ 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}.
 
-The expression inside the inner square root of \eqref{eq:energies} is
+The discriminant $\Delta_f$ inside the square root of \eqref{eq:energies} is
 proportional to
 \begin{equation} \label{eq:condition}
-  G_f
-  =
-  f'\log\frac{f''}{f'}\big[
-    3y_f(f''-f')f'''-2(f'-2f)f''w_f
-  \big]
-  -2(f''-f')u_fw_f
-  -2\log^2\frac{f''}{f'}f'^2f''v_f
+  \begin{aligned}
+    G_f
+    &=
+    f'\log\frac{f''}{f'}\big[
+      3y_f(f''-f')f'''-2(f'-2f)f''w_f
+    \big] \\
+    &\qquad-2(f''-f')u_fw_f
+    -2\log^2\frac{f''}{f'}f'^2f''v_f
+  \end{aligned}
 \end{equation}
 If $G_f>0$, then the bifurcating solutions exist, and there are some saddles whose
 complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution.
@@ -406,23 +417,23 @@ complexity. If $G_f<0$, then there is nowhere along the extremal line where
 saddles can be described by such a complexity. The range of $3+s$ models where
 $G_f$ is positive is shown in Fig.~\ref{fig:phases}.
 
-\begin{figure}
+\begin{figure*}
   \centering
-  \includegraphics{figs/range_plot_1.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_2.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_3.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_4.pdf} \\
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_1.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_2.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_3.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_4.pdf} \\
   \vspace{-2em}
-  \includegraphics{figs/range_plot_log_1.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_log_2.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_log_3.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/range_plot_log_4.pdf}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_1.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_2.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_3.pdf}
+  \hspace{-3.25em}
+  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_4.pdf}
 
   \caption{
     The range of energies where \textsc{rsb} saddles are found for the $3+s$
@@ -436,7 +447,7 @@ $G_f$ is positive is shown in Fig.~\ref{fig:phases}.
     dashed. Also marked is the range of $\lambda$ for which the ground state
     minima are characterized by nontrivial \textsc{rsb}.
   } \label{fig:energy_ranges}
-\end{figure}
+\end{figure*}
 
 Fig.~\ref{fig:energy_ranges} shows the range of energies where nontrivial
 correlations are found between stationary points in several $3+s$ models as
@@ -457,9 +468,9 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
 
 \begin{figure}
   \centering
-  \includegraphics{figs/order_plot_1.pdf}\\
+  \includegraphics[width=0.95\columnwidth]{figs/order_plot_1.pdf}\\
   \vspace{-1em}
-  \includegraphics{figs/order_plot_2.pdf}
+  \includegraphics[width=0.95\columnwidth]{figs/order_plot_2.pdf}
 
   \caption{
     Examples of $3+14$ models where the solution
@@ -486,9 +497,11 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
 There are implications for the emergence of \textsc{rsb} in equilibrium.
 Consider a specific $H$ with
 \begin{equation}
-  H(\pmb\sigma)
-  =\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p}
-  +\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s}
+  \begin{aligned}
+    H(\pmb\sigma)
+    &=\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p} \\
+    &\hspace{6pc}+\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s}
+  \end{aligned}
 \end{equation}
 where the interaction tensors $J$ are drawn from zero-mean normal distributions
 with $\overline{(J^{(p)})^2}=p!/2N^{p-1}$ and likewise for $J^{(s)}$. Functions $H$ defined this way have the covariance