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@@ -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} |