From 49934b54a919f6e10fef2f33a934958415eaf40c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 2 Feb 2022 22:01:12 +0100 Subject: Tweaked first figure, and added another depicting a Stokes point. --- figs/antithimble_homology.eps | 2 +- figs/thimble_homology.eps | 65 ++++++++++++++++-------------------------- figs/thimble_stokes_1.pdf | Bin 0 -> 17780 bytes figs/thimble_stokes_2.pdf | Bin 0 -> 19307 bytes figs/thimble_stokes_3.pdf | Bin 0 -> 19988 bytes stokes.tex | 20 +++++++++++++ 6 files changed, 45 insertions(+), 42 deletions(-) create mode 100644 figs/thimble_stokes_1.pdf create mode 100644 figs/thimble_stokes_2.pdf create mode 100644 figs/thimble_stokes_3.pdf diff --git a/figs/antithimble_homology.eps b/figs/antithimble_homology.eps index 87a534c..e1907ec 100644 --- a/figs/antithimble_homology.eps +++ b/figs/antithimble_homology.eps @@ -1,6 +1,6 @@ %!PS-Adobe-3.0 EPSF-3.0 %%Creator: cairo 1.15.4 (http://cairographics.org) -%%CreationDate: Wed Feb 2 18:57:13 2022 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a/figs/thimble_stokes_1.pdf b/figs/thimble_stokes_1.pdf new file mode 100644 index 0000000..d291173 Binary files /dev/null and b/figs/thimble_stokes_1.pdf differ diff --git a/figs/thimble_stokes_2.pdf b/figs/thimble_stokes_2.pdf new file mode 100644 index 0000000..30cdb9a Binary files /dev/null and b/figs/thimble_stokes_2.pdf differ diff --git a/figs/thimble_stokes_3.pdf b/figs/thimble_stokes_3.pdf new file mode 100644 index 0000000..d6a19c7 Binary files /dev/null and b/figs/thimble_stokes_3.pdf differ diff --git a/stokes.tex b/stokes.tex index 4c1a882..6485ed7 100644 --- a/stokes.tex +++ b/stokes.tex @@ -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 -- cgit v1.2.3-70-g09d2