From 08ba6a632cf36ff5bf16bc024770e9bf0bf282ed Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 30 Jun 2022 14:37:41 +0200
Subject: Some figure work, and some interpretation work.

---
 frsb_kac-rice.tex | 49 +++++++++++++++++++++++++++++++++++++++++++++----
 1 file changed, 45 insertions(+), 4 deletions(-)

(limited to 'frsb_kac-rice.tex')

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
@@ -512,6 +509,13 @@ E\rangle_2$.
   } \label{fig:2rsb.complexity}
 \end{figure}
 
+\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}
@@ -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 $b$:
-- 
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