From 850d60ccb2b35088c2119be387858b4e3643d507 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 10 Jul 2022 17:21:45 +0200
Subject: More 1RSB figure updating.

---
 frsb_kac-rice.tex | 29 ++++++++++++++++++++++-------
 1 file changed, 22 insertions(+), 7 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index db36220..a1ac0c1 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -883,14 +883,13 @@ lower energy than the equilibrium ground state. The 1RSB complexity resolves
 these problems, predicting the same ground state as equilibrium and that the
 complexity of marginal minima (and therefore all saddles) vanishes at
 $E_m=-1.287\,605\,527\ldots$, which is very slightly greater than $E_0$. Saddles
-become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,605\,716\ldots$.
-Finally, the 1RSB complexity transitions to a RS description at an energy
-$E_1=-1.271\,359\,96\ldots$. All these complexities can be seen plotted in
+become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,575\,114\ldots$.
+The 1RSB complexity transitions to a RS description for dominant stationary points at an energy
+$E_1=-1.273\,886\,852\ldots$. The highest energy for which the 1RSB description exists is $E_\mathrm{max}=-0.886\,029\,051\ldots$
+
+All these complexities can be seen plotted in
 Fig.~\ref{fig:2rsb.complexity}.
 
-All of the landmark energies associated with the complexity are a great deal
-smaller than their equilibrium counterparts, e.g., comparing $E_1$ and $\langle
-E\rangle_2$.
 
 
 \begin{figure}
@@ -909,8 +908,8 @@ E\rangle_2$.
 \begin{figure}
   \centering
   \includegraphics{figs/316_complexity_contour_1.pdf}
-  \hfill
   \includegraphics{figs/316_complexity_contour_2.pdf}
+  \raisebox{4em}{\includegraphics{figs/316_complexity_contour_leg.pdf}}
 
   \caption{
     Complexity of the $3+16$ model in the energy $E$ and stability $\mu$
@@ -938,6 +937,22 @@ E\rangle_2$.
   }
 \end{figure}
 
+Fig.~\ref{fig:2rsb.comparison} shows the saddle parameters for the $3+16$
+system for notable species of stationary points, notably the most common, the
+marginal ones, those with zero complexity, and those on the transition line.
+When possible, these are compared with the same expressions in the equilibrium
+solution at the same average energy. Besides the agreement at the ground state
+energy, there seems to be little correlation between the equilibrium and
+complexity parameters.
+
+Of specific note is what happens to $d_1$ as the 1RSB phase boundary for the
+complexity meets the zero complexity line. Here, $d_1$ diverges like
+\begin{equation}
+  d_1=-\left(\frac1{f'(1)}-(d_d+r_d^2)\right)(1-x_1)^{-1}+O(1)
+\end{equation}
+while $x_1$ and $q_1$ both go to one. Note that this is the only place along
+the phase boundary where $q_1$ goes to one.
+
 \begin{figure}
   \centering
   \includegraphics{figs/316_comparison_q.pdf}\hfill
-- 
cgit v1.2.3-70-g09d2

