From 601fa161fd0ca1c9570771287aa1cbcfd0603b80 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 23 Aug 2023 17:28:35 +0200
Subject: Modified article for two columns.
---
when_annealed.tex | 113 ++++++++++++++++++++++++++++++------------------------
1 file changed, 63 insertions(+), 50 deletions(-)
diff --git a/when_annealed.tex b/when_annealed.tex
index 77e07e1..0304218 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -1,4 +1,4 @@
-\documentclass{epl2/epl2}
+\documentclass[doublecol]{epl2/epl2}
\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
\usepackage[T1]{fontenc} % vector fonts
@@ -162,7 +162,7 @@ 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
@@ -216,12 +216,15 @@ 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)
@@ -254,6 +257,7 @@ 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
@@ -291,12 +295,11 @@ 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
@@ -321,13 +324,15 @@ 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
@@ -344,7 +349,7 @@ $e$, $g$, 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
@@ -365,11 +370,15 @@ 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
@@ -381,16 +390,18 @@ 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.
@@ -399,23 +410,23 @@ 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 $3+s$
@@ -429,7 +440,7 @@ $G_f$ 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
@@ -450,9 +461,9 @@ 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
@@ -479,9 +490,11 @@ 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
--
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