diff options
| -rw-r--r-- | when_annealed.tex | 113 |
1 files 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 |