From 460531efe135f92caaaf04f086ffff6da1efe1ae Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 28 Mar 2022 18:37:32 +0200
Subject: Some work on the 2-replica case.

---
 stokes.tex | 54 +++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 53 insertions(+), 1 deletion(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index f01a4b1..ed15b88 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1460,7 +1460,7 @@ in Stokes points.
 
 The problem of counting the density of Stokes points in an analytic
 continuation of the spherical models is quite challenging, as the problem of
-finding dyramic trajectories with endpoints at stationary points is already
+finding dynamic trajectories with endpoints at stationary points is already
 difficult, and once made complex the problem has twice the number of fields
 squared.
 
@@ -1596,6 +1596,58 @@ function of $\Delta$ and $\arg\delta$.
   }
 \end{figure}
 
+A lot of information is contained in the full two-replica complexity, but we
+will focus on the following question: what does the population of stationary
+points nearby a given real stationary point look like? We think this is a
+relevant question for the tendency for Stokes lines, for the following reason.
+To determine whether two given stationary points, when tuned to have the same
+imaginary energy, will share a Stokes line, one needs to solve what is known as
+the global connection problem. As we have seen, this as a question of a kind of
+adjacency: two points will \emph{not} share a Stokes line if a third intervenes
+with its thimble between them. We reason that the number of `adjacent'
+stationary points of a given stationary point for a generic function in $D$
+complex dimensions scales linearly with $D$. Therefore, if the collection of
+nearest neighbors has a finite complexity, e.g., scales \emph{exponentially}
+with $D$, crowding around the stationary point in question, then these might be
+expected to overwhelm the possible adjacencies, and so doing simplify the
+problem of determining the properties of the true adjacencies. Until the
+nonlinear flow equations are solved with dynamical mean field theory as has
+been done for instantons \cite{Ros_2021_Dynamical}, this is the best heuristic.
+
+First, we find that for all displacements $\Delta$ and real energies $\epsilon_1$, the maximum complexity is found for some real values of $\epsilon_2$ and $\delta$. Therefore we can restrict our study of the most common neighbors to this. Note that the real part of $\delta$ has a very geometric interpretation in terms of the properties of the neighbors: if a stationary point sits in the complex configuration space near another, $\operatorname{Re}\delta$ can be related to the angle $\varphi$ made between the vector separating these two points and the real configuration space as
+\begin{equation}
+  \varphi=\arctan\sqrt{\frac{1+\operatorname{Re}\delta}{1-\operatorname{Re}\delta}}
+\end{equation}
+Having concluded that the most populous neighbors are confined to real $\delta$, we will make use of this angle instead of $\delta$, which has a more direct geometric interpretation.
+
+\begin{figure}
+  \hspace{5pc}
+  \includegraphics{figs/neighbor_geometry.pdf}
+
+  \caption{
+    The geometric definition of the angle $\varphi$, between the displacement
+    between two stationary points and the real configuration space.
+  }
+\end{figure}
+
+First, we examine the important of the threshold.
+Fig.~\ref{fig:neighbor.complexity.passing.threshold} shows the two-replica
+complexity evaluated at $\Delta=2^{-4}$ and equal energy
+$\epsilon_2=\epsilon_1$ as a function of $\varphi$ for several $\epsilon_1$ as
+the threshold is passed. The curves are rescaled by the complexity
+$\Sigma_2(\epsilon_1)$ of index 2 saddles in the real problem, which is what is
+approached in the limit as $\Delta$ to zero. Below the threshold, the
+distribution of nearby saddles with the same energy by angle is broad and
+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.
+
+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 $\Delta^3$.
+
+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.
+
 \subsection{Pure {\it p}-spin: is analytic continuation possible?}
 
 \begin{equation}
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