From 0cc08de09123f0e73b6087a053f9ec249c933b37 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 28 May 2023 11:11:30 +0200
Subject: Added discussion of the stability and the models of choice.

---
 2-point.tex | 47 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 47 insertions(+)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index af0af87..d39f12c 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -159,7 +159,54 @@ The gradient and Hessian at a stationary point are then
 \end{align}
 where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivative with respect to $\mathbf s$.
 
+When we count stationary points, we classify them by certain properties. One of
+these is the energy density $E=H/N$. We will also fix the \emph{stability}
+$\mu=\frac1N\operatorname{Tr}\operatorname{Hess}H$, also known as the radial
+reaction. In the mixed spherical models, all stationary points have a
+semicircle law for the eigenvalue spectrum of their Hessians, each with the
+same width $\mu_\mathrm m$, but whose center is shifted by different amounts. Fixing the
+stability $\mu$ fixes this shift, and therefore fixes the spectrum of the
+associated stationary point. When the stability is smaller than the width of
+the spectrum, or $\mu<\mu_\mathrm m$, there are an extensive number of negative
+eigenvalues, and the stationary point is a saddle with same large index whose
+value is set by the stability. When the stability is greater than the width of
+the spectrum, or $\mu>\mu_\mathrm m$, the semicircle distribution lies only
+over positive eigenvalues, and unless an isolated eigenvalue leaves the
+semicircle and becomes negative, the stationary point is a minimum. Finally,
+when $\mu=\mu_\mathrm m$, the edge of the semicircle touches zero and we have
+marginal minima.
+
+In the pure spherical models, $E$ and $\mu$ cannot be fixed separately: fixing
+one uniquely fixes the other. This property leads to the great simplification
+of these models: marginal minima exist \emph{only} at one energy level, and
+therefore only that energy has the possibility of trapping the long-time
+dynamics.
+
+\subsection{Models of focus}
+
+In this study, we focus exclusively on models whose complexity is replica symmetric. We study two models of interest, both with concave $f''(q)^{-1/2}$: a $3+4$ model whose dynamics were studied extensively in \cite{Folena_2020_Rethinking}, given by
+\begin{equation}
+  f_{3+4}(q)=\frac12\big(q^3+q^4\big)
+\end{equation}
+and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics recently studied in \cite{Folena_2023_On} given by
+\begin{equation}
+  f_{3+8}(q)=\frac12\big(\tfrac{76}{100}q^3+\tfrac{24}{100}q^8\big)
+\end{equation}
+
+\begin{figure}
+  \caption{
+    Plots of the complexity (logarithm of the number of stationary points) for
+    the mixed spherical models studied in this paper. Energies and stabilities
+    of interest are marked, including the ground state energy and stability
+    $E_\mathrm{gs}$ and $\mu_\mathrm{gs}$, the marginal stability $\mu_\mathrm
+    m$, and the threshold energy $E_\mathrm{th}$. The line shows the location
+    of the most common type of stationary point at each energy level. Estimated
+    locations of notable attractors of the dynamics are highlighted.
+  } \label{fig:complexities}
+\end{figure}
+
 \section{Results}
+\label{sec:results}
 
 \begin{figure}
   \includegraphics{figs/gapped_min_energy.pdf}
-- 
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