From f9fa2d8bbc59825086358d2cef116ec5af57d503 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 15 Jun 2022 10:11:14 +0200
Subject: Lots of figures and writing.

---
 frsb_kac_new.tex | 136 ++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 120 insertions(+), 16 deletions(-)

(limited to 'frsb_kac_new.tex')

diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex
index e8b57c8..dcf6dd8 100644
--- a/frsb_kac_new.tex
+++ b/frsb_kac_new.tex
@@ -455,22 +455,13 @@ When minima dominate, $\mu>\mu_m$ and all the roots inside $\mathcal D(\mu)$ are
   \mu=\frac1{R_d}+R_df''(1)
 \end{equation}
 
-\begin{figure}
-  \centering
-  \includegraphics{figs/24_complexity.pdf}
-  \caption{
-    The complexity $\Sigma$ of the mixed $2+4$ spin model as a function of
-    distance $\Delta\epsilon=\epsilon-\epsilon_0$ of the ground state. The
-    solid blue line shows the complexity of dominant saddles given by the FRSB
-    ansatz, and the solid yellow line shows the complexity of marginal minima.
-    The dashed lines show the same for the annealed complexity. The inset shows
-    more detail around the ground state.
-  } \label{fig:frsb.complexity}
-\end{figure}
+\subsubsection{Recovering the equilibrium ground state}
 
-\subsubsection{Recovering the replica ground state}
-
-The ground state energy corresponds to that where the complexity of dominant stationary points becomes zero. If the most common stationary points vanish, then there cannot be any stationary points. In this section, we will show that it reproduces the ground state produced by taking the zero-temperature limit in the equilibrium case.
+The ground state energy corresponds to that where the complexity of dominant
+stationary points becomes zero. If the most common stationary points vanish,
+then there cannot be any stationary points. In this section, we will show that
+it reproduces the ground state produced by taking the zero-temperature limit in
+the equilibrium case.
 
 Consider the extremum problem of \eqref{eq:diagonal.action} with respect to $R_d$ and $D_d$. This gives the equations
 \begin{align}
@@ -526,7 +517,120 @@ complexity in the ground state are
   D_d=\hat\beta R_d
 \end{align}
 
-\subsection{The continuum situation at a glance}
+\section{Examples}
+
+\subsection{1RSB complexity}
+
+It is known that by choosing a covariance $f$ as the sum of polynomials with
+well-separated powers, one develops 2RSB in equilibrium. This should correspond
+to 1RSB in Kac--Rice. For this example, we take
+\begin{equation}
+  f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
+\end{equation}
+With this covariance, the model sees a RS to 1RSB transition at
+$\beta_1=1.70615\ldots$ and a 1RSB to 2RSB transition at $\beta_2=6.02198\ldots$. At these points, the average energies are $\langle E\rangle_1=-0.906391\ldots$ and $\langle E\rangle_2=-1.19553\ldots$, and the ground state energy is $E_0=-1.2876055305\ldots$.
+
+In this model, the RS complexity gives an inconsistent answer for the
+complexity of the ground state, predicting that the complexity of minima
+vanishes at a higher energy than the complexity of saddles, with both at a
+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
+$E_m=-1.2876055265\ldots$, which is very slightly greater than $E_0$. Saddles
+become dominant over minima at a higher energy $E_s=-1.287605716\ldots$.
+Finally, the 1RSB complexity transitions to a RS description at an energy
+$E_1=-1.27135996\ldots$. 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}
+  \centering
+  \includegraphics{figs/316_complexity.pdf}
+
+  \caption{
+    Complexity of dominant saddles (blue), marginal minima (yellow), and
+    dominant minima (green) of the $3+16$ model. Solid lines show the result of
+    the 1RSB ansatz, while the dashed lines show that of a RS ansatz. The
+    complexity of marginal minima is always below that of dominant critical
+    points except at the red dot, where they are dominant.
+    The inset shows a region around the ground state and the fate of the RS solution.
+  } \label{fig:2rsb.complexity}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \begin{minipage}{0.7\textwidth}
+    \includegraphics{figs/316_comparison_q.pdf}
+    \hspace{1em}
+    \includegraphics{figs/316_comparison_x.pdf} \vspace{1em}\\
+    \includegraphics{figs/316_comparison_b.pdf}
+    \hspace{1em}
+    \includegraphics{figs/316_comparison_R.pdf}
+  \end{minipage}
+  \includegraphics{figs/316_comparison_legend.pdf}
+
+  \caption{
+    Comparisons between the saddle parameters of the equilibrium solution to
+    the $3+16$ model (blue) and those of the complexity (yellow). Equilibrium
+    parameters are plotted as functions of the average energy $\langle
+    E\rangle=\partial_\beta(\beta F)$ and complexity parameters as functions of
+    fixed energy $E$. Solid lines show the result of a 2RSB ansatz, dashed
+    lines that of a 1RSB ansatz, and dotted lines that of a RS ansatz. All
+    paired parameters coincide at the ground state energy, as expected.
+  } \label{fig:2rsb.comparison}
+\end{figure}
+
+\subsection{Full RSB complexity}
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/24_complexity.pdf}
+  \caption{
+    The complexity $\Sigma$ of the mixed $2+4$ spin model as a function of
+    distance $\Delta\epsilon=\epsilon-\epsilon_0$ of the ground state. The
+    solid blue line shows the complexity of dominant saddles given by the FRSB
+    ansatz, and the solid yellow line shows the complexity of marginal minima.
+    The dashed lines show the same for the annealed complexity. The inset shows
+    more detail around the ground state.
+  } \label{fig:frsb.complexity}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/24_func.pdf}
+  \hspace{1em}
+  \includegraphics{figs/24_qmax.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}
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/24_comparison_b.pdf}
+  \hspace{1em}
+  \includegraphics{figs/24_comparison_Rd.pdf}
+  \raisebox{3em}{\includegraphics{figs/24_comparison_legend.pdf}}
+
+  \caption{
+    Comparisons between the saddle parameters of the equilibrium solution to
+    the $3+4$ model (black) and those of the complexity (blue and yellow). Equilibrium
+    parameters are plotted as functions of the average energy $\langle
+    E\rangle=\partial_\beta(\beta F)$ and complexity parameters as functions of
+    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}
+\end{figure}
 
 In the case where any FRSB is present, one must work with the functional form
 of the complexity \eqref{eq:functional.action}, which must be extremized with
-- 
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