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@@ -257,6 +257,26 @@ called \emph{Stokes points}, and the gradient descent trajectories that join two stationary points are called \emph{Stokes lines}. An example of this behavior can be seen in Fig.~\ref{fig:1d.stokes}. +\begin{figure} + \includegraphics{figs/thimble_stokes_1.pdf}\hfill + \includegraphics{figs/thimble_stokes_2.pdf}\hfill + \includegraphics{figs/thimble_stokes_3.pdf} + + \caption{ + An example of a Stokes point. (Left) The collection of thimbles necessary + to progress around from left to right, highlighted in a darker color, is + the same as it was in Fig.~\ref{fig:thimble.homology}. (Center) The thimble + $\mathcal J_\blacklozenge$ intersects the stationary point $\blacktriangle$ + and its thimble, leading to a situation where the contour is not easily + defined using thimbles. This is a Stokes point. (Right) The Stokes point + has passed, and the collection of thimbles necessary to produce the path + has changed: now $\mathcal J_\blacktriangle$ must be included. Notice that + in this figure, because of the symmetry of the pure models, the thimble + $\mathcal J_\blacksquare$ also experiences a Stokes point, but this does + not result in a change to the path involving that thimble. + } \label{fig:1d.stokes} +\end{figure} + The prevalence (or not) of Stokes points in a given continuation, and whether those that do appear affect the weights of critical points of interest, is a concern for the analytic continuation of theories. If they do not occur or |