From d5cee36e3478b13ef0b0ac3cee18d88cf4d7786e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 28 Mar 2022 23:54:09 +0200
Subject: Added a lot of new figures and writing to the two-spin section.

---
 stokes.tex | 85 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 84 insertions(+), 1 deletion(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index ed15b88..1c60a0d 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1642,11 +1642,94 @@ peaked around $\varphi=45^\circ$, while above the threshold it is peaked
 strongly near the minimum allowed $\varphi$. At the threshold, the function
 becomes extremely flat.
 
+\begin{figure}
+  \includegraphics{figs/neighbor_thres.pdf}
+  \caption{
+    The scaled two-replica complexity $\Upsilon$ as a function of angle
+    $\varphi$ with $\epsilon_2=\epsilon_1$, $\Delta=2^{-7}$, and various
+    $\epsilon_1$. At the threshold, the function undergoes a geometric
+    transition and becomes sharper with decreasing $\Delta$.
+  } \label{fig:neighbor.complexity.passing.threshold}
+\end{figure}
+
 One can examine the scaling of these curves as $\Delta$ goes to zero. Both above and below the threshold, one finds a quickly-converging limit of $(\Sigma(\epsilon_1,\epsilon_1,\varphi,\Delta)/\Sigma_2(\epsilon_1)-1)/\Delta$. Above the threshold, these curves converge to a function whose peak is always precisely at $45^\circ$, while below they converge to a function with a peak that grows linearly with $\Delta^{-1}$. At the threshold, the scaling is different, and the function approaches a flat function extremely rapidly, as $\Delta^3$.
 
+\begin{figure}
+  \includegraphics{figs/neighbor_limit_thres_above.pdf}
+  \hspace{-1em}
+  \includegraphics{figs/neighbor_limit_thres_at.pdf}
+  \hspace{-1em}
+  \includegraphics{figs/neighbor_limit_thres_below.pdf}
+  \hfill
+  \includegraphics{figs/neighbor_limit_thres_legend.pdf}
+  \caption{
+    The scaled two-replica complexity $\Upsilon$ as a function of angle
+    $\varphi$ for various $\Delta$, $\epsilon_2=\epsilon_1$, and \textbf{Left:}
+    $\epsilon_1=\epsilon_\mathrm{th}+0.001$ \textbf{Center:}
+    $\epsilon_1=\epsilon_\mathrm{th}$ \textbf{Right:}
+    $\epsilon=\epsilon_\mathrm{th}-0.001$. All lines have been normalized by
+    the complexity $\Sigma$ of the real 3-spin model at the same energy, or
+    (where relevant) by the complexity $\Sigma_2$ of rank-two saddles of the
+    real 3-spin model.
+  }
+\end{figure}
+
 Thus, there is an abrupt geometric transition in the population of nearest
 neighbors as the threshold is crossed: above they are broadly distributed at
-all angles, while below they are highly concentrated around $90^\circ$. From this analysis it appears that the complexity of the nearest neighbors, at zero distance, behaves as that of the index-2 saddles at all angles, which would imply that the nearest neighbors vanish at the same point as the index-2 saddles. However, this is not the case: we have only shown that this is how the neighbors at \emph{identical energy} scale, which is correct above the threshold, but no longer underneath.
+all angles, while below they are highly concentrated around $90^\circ$. From
+this analysis it appears that the complexity of the nearest neighbors, at zero
+distance, behaves as that of the index-2 saddles at all angles, which would
+imply that the nearest neighbors vanish at the same point as the index-2
+saddles. However, this is not the case: we have only shown that this is how the
+neighbors at \emph{identical energy} scale, which is correct above the
+threshold, but no longer underneath.
+
+If an energy is taken under the threshold and the two-replica complexity
+maximized with respect to both $\epsilon_2$ and $\varphi$, one finds that as
+$\Delta\to0$, $\epsilon_2\to\epsilon_1$, as must be the case the find a
+positive complexity at zero distance, but the maximum is never at
+$\epsilon_2=\epsilon_1$, but rather at a small distance $\Delta\epsilon$ that
+decreases with decreasing $\Delta$ like $\Delta^2$. When the complexity is
+maximized in both parameters, one finds that, in the limit as $\Delta\to0$, the
+peak is at $90^\circ$ but has a height equal to $\Sigma_1$, the complexity of
+rank-1 saddles.
+
+\begin{figure}
+  \includegraphics{figs/neighbor_energy_limit.pdf}
+  \caption{
+    The two-replica complexity $\Upsilon$ scaled by $\Sigma_1$ as a function of
+    angle $\varphi$ for various $\Delta$ at $\epsilon_1=\mathcal E_2$, the
+    point of zero complexity for rank-two saddles in the real problem.
+    \textbf{Solid lines:} The complexity evaluated at the value of $\epsilon_2$
+    which leads to the largest maximum value. As $\Delta$ varies this varies
+    like $\epsilon_2-\epsilon_1\propto\Delta^2$. \textbf{Dashed lines:} The
+    complexity evaluated at $\epsilon_2=\epsilon_1$.
+  }
+\end{figure}
+
+Below $\mathcal E_1$, where the rank-1 saddle complexity vanishes, the complexity of stationary points of any type at zero distance is negative. To find what the nearest population looks like, one must find the minimum $\Delta$ at which the complexity is nonnegative, or
+\begin{equation}
+  \Delta_\textrm{min}=\operatorname{argmin}_\Delta\left(0\leq\max_{\epsilon_2, \varphi}\Upsilon(\epsilon_1,\epsilon_2,\Delta,\varphi)\right)
+\end{equation}
+The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produces it is plotted in Fig.~\ref{fig:nearest.properties}. As the energy is brought below $\mathcal E_1$, $\epsilon_2-\epsilon_1\propto -|\epsilon_1-\mathcal E_1|^2$, $\varphi-90^\circ\propto-|\epsilon_1-\mathcal E_1|^{1/2}$, and $\Delta_\textrm{min}\propto|\epsilon_1-\mathcal E_1|$. The fact that the population of nearest neighbors has a energy lower than the stationary point gives some hope for the success of continuation involving these points: since Stokes points only lead to a change in weight when they involve upward flow from a point that already has weight, neighbors that have a lower energy won't be eligible to be involved in a Stokes line that causes a change of weight until the phase of $\beta$ has rotated almost $180^\circ$.
+
+\begin{figure}
+  \includegraphics{figs/neighbor_plot.pdf}
+
+  \caption{
+    The properties of the nearest neighbor saddles as a function of energy
+    $\epsilon$. Above the threshold energy $\mathcal E_\mathrm{th}$, stationary
+    points are found at arbitrarily close distance and at all angles $\varphi$
+    in the complex plane. Below $\mathcal E_\mathrm{th}$ but above $\mathcal
+    E_2$, stationary points are still found at arbitrarily close distance and
+    all angles, but there are exponentially more found at $90^\circ$ than at
+    any other angle. Below $\mathcal E_2$ but above $\mathcal E_1$, stationary
+    points are found at arbitrarily close distance but only at $90^\circ$.
+    Below $\mathcal E_1$, neighboring stationary points are separated by a
+    minimum squared distance $\Delta_\textrm{min}$, and the angle they are
+    found at drifts.
+  } \label{fig:nearest.properties}
+\end{figure}
 
 \subsection{Pure {\it p}-spin: is analytic continuation possible?}
 
-- 
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