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@@ -635,7 +635,7 @@ It therefore follows that the eigenvalues and vectors of the real hessian satisf \beta\operatorname{Hess}\mathcal S v=\lambda v^* \end{equation} a sort of generalized -eigenvalue problem. If we did not know the eigenvalues were real, we could +eigenvalue problem, whose solutions are called \emph{Takagi values and vectors} \cite{Takagi_1924_On}. If we did not know the eigenvalues were real, we could still see it from the second implied equation, $(\beta\operatorname{Hess}\mathcal S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. @@ -1801,6 +1801,50 @@ two stationary points. Among minimized lines these values fall into doubly-peaked histograms which well-separated prospective Stokes lines into `good' and `bad' values for the given level of approximation $m$. +One cannot explicitly study the effect of crossing various landmark energies on +the $p$-spin in the system sizes that were accessible to our study, up to +around $N=64$, as the presence of, e.g., the threshold energy separating +saddles from minima is not noticeable until much larger size +\cite{Folena_2020_Rethinking}. However, we are +able to examine the effect of its symptoms: namely, the influence of the +spectrum of the stationary point in question on the likelihood that a randomly +chosen neighbor will share a Stokes line. + +\begin{figure} + \includegraphics{figs/numerics_prob_eigenvalue.pdf} + + \caption{ + The probability $P_\mathrm{Stokes}$ that a real stationary point will share + a Stokes line with its randomly chosen neighbor as a function of + $|\lambda_\textrm{min}|$, the magnitude of the minimum eigenvalue of the + hessian at the real stationary point. The horizontal axis has been rescaled + to collapse the data at different system sizes $N$. + } \label{fig:numeric.prob.eigenvalue} +\end{figure} + +Data for the likelihood of a Stokes line as a function of the empirical gap +$|\lambda_\textrm{min}|$ of the real stationary point is shown in +Fig.~\ref{fig:numeric.prob.eigenvalue}. There, one sees that the probability of +finding a Stokes line with a near neighbor falls off as an exponential in the +magnitude of the smallest eigenvalue. As a function of system size, the tail +contracts like $N^{-1/2}$, which means that in the thermodynamic limit one +expects the probability of finding such a Stokes line will approach zero +everywhere expect where $\lambda_\textrm{min}\ll1$. This supports the idea that +gapped minima are unlikely to see Stokes lines. + +\begin{figure} + \includegraphics{figs/numerics_angle_gap_32.pdf} + + \caption{ + The probability density function for identified Stokes points as a function + of $|\theta|$, the magnitude of the phase necessary to add to $\beta$ to + reach the Stokes point, at $N=32$ and for several binned + $|\lambda_\textrm{min}|$. As the empirical gap is increased, the population + of discovered Stokes points becomes more concentrated around + $|\theta|=\pi$. + } \label{fig:numeric.angle.gap} +\end{figure} + \subsection{Pure {\it p}-spin: is analytic continuation possible?} After this work, one is motivated to ask: can analytic continuation be done in |