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| -rw-r--r-- | figs/316_complexity.pdf | bin | 33069 -> 26267 bytes | |||
| -rw-r--r-- | frsb_kac-rice.tex | 49 |
2 files changed, 45 insertions, 4 deletions
diff --git a/figs/316_complexity.pdf b/figs/316_complexity.pdf Binary files differindex b656641..6bb59f3 100644 --- a/figs/316_complexity.pdf +++ b/figs/316_complexity.pdf diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 0896da4..180798a 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -497,10 +497,7 @@ E\rangle_2$. \begin{figure} - \centering - \includegraphics{figs/316_complexity_contour_1.pdf} - \hfill - \includegraphics{figs/316_complexity_contour_2.pdf} + \includegraphics{figs/316_complexity.pdf} \caption{ Complexity of dominant saddles (blue), marginal minima (yellow), and @@ -514,6 +511,13 @@ E\rangle_2$. \begin{figure} \centering + \includegraphics{figs/316_complexity_contour_1.pdf} + \hfill + \includegraphics{figs/316_complexity_contour_2.pdf} +\end{figure} + +\begin{figure} + \centering \begin{minipage}{0.7\textwidth} \includegraphics{figs/316_comparison_q.pdf} \hspace{1em} @@ -633,6 +637,43 @@ for different energies and typical vs minima. \section{Interpretation} +\begin{equation} + H(s)-h^Ts+g\xi^Ts +\end{equation} +Let $\langle A\mid\epsilon,\mu\rangle$ be average over stationary points with given $\epsilon$ and $\mu$, i.e., +\begin{equation} + \langle A\mid\epsilon,\mu\rangle + =\frac1{\mathcal N} + \int d\nu(s\mid\epsilon,\mu)\,A(s) +\end{equation} +with +\begin{equation} + d\nu(s\mid\epsilon,\mu)=ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)| +\end{equation} +\begin{equation} + \frac1N\|\langle s\mid\epsilon,\mu\rangle\|^2 + =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha\mid\epsilon,\mu)\,\left(\frac1{n(n-1)}\sum_{\alpha\neq\beta}\frac{s_\alpha^Ts_\beta}N\right) + =\lim_{n\to0}\frac1{n(n-1)}\left\langle\sum_{a\neq b}^nC_{ab}\right\rangle + =\int_0^1 dx\,c(x) +\end{equation} + +\begin{equation} + \frac1N\sum_i\frac{\partial\langle s_i\rangle}{\partial h_i} + =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha)\,\left(\frac1n\sum_{\alpha\beta}-i\frac{\hat s_\alpha^Ts_\beta}N\right) + =\lim_{n\to0}\frac1n\left\langle\sum_{\alpha\beta}R_{\alpha\beta}\right\rangle + =r_d-\int_0^1dx\,r(x) +\end{equation} + +\begin{equation} + \begin{aligned} + \lim_{g\to0}\overline{\frac{\partial^2\Sigma}{\partial g^2}} + =\frac1N\lim_{g\to0}\lim_{n\to0}\frac1n\overline{\int\prod_\alpha d\nu(s_\alpha)\left(\sum_\alpha i\xi^T\hat s_\alpha\right)^2} + =\lim_{n\to0}\frac1n\int\prod_\alpha d\nu(s_\alpha)\left(\sum_{ab}-\frac{\hat s_a^T\hat s_b}N\right) \\ + =-\lim_{n\to0}\frac1n\left\langle\sum_{ab}D_{ab}\right\rangle + =-d_d+\int_0^1dx\,d(x) + \end{aligned} +\end{equation} + The meaning of $R_{ab}$ is that of a response of replica $a$ to a linear field in replica $b$: |