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| -rw-r--r-- | figs/order_plot_1.pdf | bin | 0 -> 11653 bytes | |||
| -rw-r--r-- | figs/order_plot_2.pdf | bin | 0 -> 14270 bytes | |||
| -rw-r--r-- | when_annealed.tex | 73 |
3 files changed, 51 insertions, 22 deletions
diff --git a/figs/order_plot_1.pdf b/figs/order_plot_1.pdf Binary files differnew file mode 100644 index 0000000..dc749d8 --- /dev/null +++ b/figs/order_plot_1.pdf diff --git a/figs/order_plot_2.pdf b/figs/order_plot_2.pdf Binary files differnew file mode 100644 index 0000000..2c2a9b3 --- /dev/null +++ b/figs/order_plot_2.pdf diff --git a/when_annealed.tex b/when_annealed.tex index 90dc067..b92e9a5 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -25,7 +25,7 @@ \begin{document} \title{ - When is the average number of saddles typical? + When is the average number of saddles in a function typical? } \author{Jaron Kent-Dobias} @@ -150,11 +150,15 @@ When the complexity is calculated using the Kac--Rice formula and a physicists' 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}. An ansatz for the complexity needs to be made. Here we restrict ourselves to a -{\oldstylenums1}\textsc{rsb} ansatz. We are guaranteed that +{\oldstylenums1}\textsc{rsb} ansatz, which is parameterized by two quantities: +$q_1$ and $x$. They have a geometric interpretation: given a stationary point +fixed with certain properties, $1-x$ corresponds to the proportion of other +stationary points with the same properties that are correlated with it, +and $q_1$ gives the overlap that this correlated population has with it. In the annealed or replica-symmetric case, $x=1$ and all but a vanishing fraction of stationary points are uncorrelated with each other. +We are guaranteed that $\Sigma\leq\Sigma_{\oldstylenums1\textsc{rsb}}\leq\Sigma_\mathrm a$, and we -will discuss later in what settings the {\oldstylenums1}\textsc{rsb} -complexity is correct. The complexity is given by extremizing an effective -action, +will discuss later in what settings the {\oldstylenums1}\textsc{rsb} complexity +is correct. 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(q_1,x\mid E,\mu)e^{nN\mathcal S(q_1,x\mid E,\mu)} =\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S(q_1,x\mid E,\mu) @@ -230,12 +234,16 @@ where we define for brevity (here and elsewhere) the constants u_f=f(f'+f'')-f'^2 && v_f=f'(f''+f''')-f''^2 \\ - w_f=2f''^2+f'(f'''-2f'') + w_f=2f''(f''-f')+f'f''' && y_f=f'(f'-f)+f''f && z_f=f(f''-f')+f'^2 \end{align} +Note that for $f$ to define a sensible covariance, all of its series +coefficients must be nonnegative. If $f$ has at least two nonzero coefficients +at second order or higher, all of these constants are positive. + Let $M$ be the matrix of double partial derivatives of $\mathcal S$ with respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point $x=1$ with the additional constraint that $q_1=1$ and along the extremal @@ -243,14 +251,14 @@ complexity line \eqref{eq:extremal.line}. We determine when a zero eigenvalue appears, indicated the presence of a bifurcating {\oldstylenums1}\textsc{rsb} solution, by solving $0=\det M$. We find \begin{equation} - \det M=-\left(\frac{\partial^2\mathcal S}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\right)^2\propto(ay^2+bE^2+2cyE+d)^2 + \det M=-\left(\frac{\partial^2\mathcal S}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\right)^2\propto(ay^2+bE^2+2cyE-d)^2 \end{equation} 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 below. Changing to $y$ is a convenient choice because the branch of \eqref{eq:extremal.line} is chosen by the sign of $y$ (the lower-energy branch we are interested in corresponds with $y>0$) and the relationship between $y$ and $E$ on the extremal line is -$g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by +$g=2hy^2+eE^2$, where the constants $e$, $g$, and $h$ are given by \begin{equation} \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'} @@ -259,7 +267,7 @@ $g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by \\ c&=\frac{w_f}{f''z_f^2} \qquad - d=-\frac{w_f}{f'f''} + d=\frac{w_f}{f'f''} \qquad e=f''-f' \qquad @@ -271,24 +279,20 @@ $g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by The solutions for $\det M=0$ correspond to energies that satisfy \begin{equation} E_{\oldstylenums1\textsc{rsb}}^\pm - =-\operatorname{sign}[c(de+bg)]\frac{|c|g\mp\sqrt{c^2g^2-(2dh+ag)(de+bg)}} + =\operatorname{sign}(bg-de)\frac{-cg\pm\sqrt{c^2g^2+(2dh-ag)(bg-de)}} { - \sqrt{2c^2eg+(2bh-ae)(de+bg)\mp2|c|e\sqrt{c^2g^2-(2dh+ag)(de+bg)}} + \sqrt{2c^2eg+(2bh-ae)(bg-de)\mp2ce\sqrt{c^2g^2+(2dh-ag)(bg-de)}} } \end{equation} The expression inside the inner square root is proportional to \begin{equation} - \begin{aligned} - G_f - &= - -2(f''-f')u_fw_f - -2\log^2\frac{f''}{f'}f'^2f''v_f - \\ - &\qquad - -f'\log\frac{f''}{f'}\Big[ - 4(f'-2f)(f''-f')f''^2-\big(-3(f'-f)f'^2+f'(f'-2f)f''+3ff''^2\big)f''' - \Big] - \end{aligned} + 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 \end{equation} If $G_f>0$, then there are two points along the extremal complexity line where a solution bifurcates, and a new line of {\oldstylenums1}\textsc{rsb} solutions @@ -343,6 +347,31 @@ between them. Therefore, $G_f>0$ is a necessary condition to see } \end{figure} +\begin{figure} + \centering + \includegraphics{figs/order_plot_1.pdf}\\ + \vspace{-1em} + \includegraphics{figs/order_plot_2.pdf} + + \caption{ + Examples of $3+14$ models where the solution + $E_{\oldstylenums1\textsc{rsb}}^-$ does and doesn't define the lower limit + of energies where \textsc{rsb} saddles are found. In both plots the red dot + shows $E_{\oldstylenums1\textsc{rsb}}^-$, while the solid red lines shows + the transition boundary with the \textsc{rs} complexity. The dashed black + line shows the \textsc{rs} zero complexity line, while the solid black line + shows the {\oldstylenums1}\textsc{rsb} zero complexity line. \textbf{Top:} + $\lambda=0.67$. Here the end of the transition line that begins at + $E_{\oldstylenums1\textsc{rsb}}^+$ does not match + $E_{\oldstylenums1\textsc{rsb}}^-$ but terminates at higher energies. + $E_{\oldstylenums1\textsc{rsb}}^-$ still corresponds with the lower bound. + \textbf{Bottom:} $\lambda=0.69$. Here the end of the transition line that + begins at $E_{\oldstylenums1\textsc{rsb}}^+$ terminates at lower energies + than $E_{\oldstylenums1\textsc{rsb}}^-$, and therefore its terminus defines + the lower bound. + } +\end{figure} + \begin{equation} \mu =-\frac{(f_1'+f_0'')u_f}{(2f_1-f_1')f_1'f_0''^{1/2}} |