From 1cf4f8a367c55b5fef42f49af5538c7812a92920 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 11 Jul 2022 17:47:26 +0200
Subject: Lots of figure work.

---
 frsb_kac-rice.tex | 44 ++++++++++++++++++++++++++++++++++----------
 1 file changed, 34 insertions(+), 10 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 130b7e9..2e0b971 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -664,7 +664,7 @@ is given by a Parisi matrix with parameters $x_1,\ldots,x_k$ and
 $q_1,\ldots,q_k$, then the parameters $\hat\beta$, $r_d$, $d_d$, $\tilde
 x_1,\ldots,\tilde x_{k-1}$, and $\tilde c_1,\ldots,\tilde c_{k-1}$ for the
 complexity in the ground state are
-\begin{align}
+\begin{align}\label{eq:equilibrium.complexity.map}
   \hat\beta=\lim_{\beta\to\infty}\beta x_k
   &&
   \tilde x_i=\lim_{\beta\to\infty}\frac{x_i}{x_k}
@@ -846,6 +846,22 @@ that this is the line of stability for the replica symmetric saddle.
 \section{General solution: examples}
 \label{sec:examples}
 
+Though we have only written down an easily computable complexity along a
+specific (and often uninteresting) line in energy and stability, this
+computable (supersymmetric) solution gives a numeric foothold for computing the
+complexity in the rest of that space. First,
+\eqref{eq:ground.state.free.energy.cont} is \emph{maximized} with respect
+to its parameters, since the equilibrium solution is equivalent to a
+variational problem. Second, the mapping \eqref{eq:equilibrium.complexity.map}
+is used to find the corresponding Kac--Rice saddle parameters in the ground
+state. With these parameters in hand, small steps are then made in energy $E$
+or stability $\mu$, after which known values are used as the initial condition
+for a saddle-finding problem. In this section, we use this basic numeric idea
+to map out the complexity for two representative examples: a model with a
+$2RSB$ equilibrium ground state and therefore $1RSB$ complexity in its
+vicinity, and a model with a $FRSB$ equilibrium ground state, and therefore
+$FRSB$ complexity as well.
+
 \subsection{1RSB complexity}
 
 It is known that by choosing a covariance $f$ as the sum of polynomials with
@@ -1002,17 +1018,22 @@ the phase boundary where $q_1$ goes to one.
 
 \begin{figure}
   \centering
-  \includegraphics{figs/24_func.pdf}
+  \includegraphics{figs/24_phases.pdf}
+  \caption{
+  } \label{fig:frsb.phases}
+\end{figure}
+
+\begin{figure}
+  \raggedright
+  \hspace{2em}\includegraphics{figs/24_opt_q(x).pdf}
+  \hspace{1em}
+  \includegraphics{figs/24_opt_xMax.pdf}
+  \\\vspace{1em}
+  \includegraphics{figs/24_opt_r(x).pdf}
   \hspace{1em}
-  \includegraphics{figs/24_qmax.pdf}
+  \includegraphics{figs/24_opt_d(x).pdf}
 
   \caption{
-    \textbf{Left:} The spectrum $\chi$ of the replica matrix in the complexity
-    of dominant saddles for the $2+4$ model at several energies.
-    \textbf{Right:} The cutoff $q_{\mathrm{max}}$ for the nonlinear part of the
-    spectrum as a function of energy $E$ for both dominant saddles and marginal
-    minima. The colored vertical lines show the energies that correspond to the
-    curves on the left.
   } \label{fig:24.func}
 \end{figure}
 
@@ -1031,7 +1052,7 @@ the phase boundary where $q_1$ goes to one.
     fixed energy $E$. Solid lines show the result of a FRSB ansatz and dashed
     lines that of a RS ansatz. All paired parameters coincide at the ground
     state energy, as expected.
-  } \label{fig:2rsb.comparison}
+  } \label{fig:frsb.comparison}
 \end{figure}
 
 
@@ -1246,6 +1267,9 @@ extracted in all detail.
 A first and very important application of the method here is to perform the calculation for high dimensional spheres, where it would give us
 a clear understanding of what happens in a low-temperature realistic jamming dynamics \cite{Maimbourg_2016_Solution}. 
 
+\paragraph{Acknowledgements}
+J K-D and J K would like to thank Valentina Ros for helpful discussions.
+
 \paragraph{Funding information}
 J K-D and J K are supported by the Simons Foundation Grant No. 454943.
 
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