diff options
| -rw-r--r-- | figs/ground_complexity.pdf | bin | 0 -> 126731 bytes | |||
| -rw-r--r-- | figs/im_complexity.pdf | bin | 0 -> 209597 bytes | |||
| -rw-r--r-- | figs/leg_complexity.pdf | bin | 0 -> 6424 bytes | |||
| -rw-r--r-- | figs/re_complexity.pdf | bin | 0 -> 109687 bytes | |||
| -rw-r--r-- | stokes.bib | 14 | ||||
| -rw-r--r-- | stokes.tex | 37 |
6 files changed, 50 insertions, 1 deletions
diff --git a/figs/ground_complexity.pdf b/figs/ground_complexity.pdf Binary files differnew file mode 100644 index 0000000..fe3a3b0 --- /dev/null +++ b/figs/ground_complexity.pdf diff --git a/figs/im_complexity.pdf b/figs/im_complexity.pdf Binary files differnew file mode 100644 index 0000000..8c5bf5b --- /dev/null +++ b/figs/im_complexity.pdf diff --git a/figs/leg_complexity.pdf b/figs/leg_complexity.pdf Binary files differnew file mode 100644 index 0000000..5b093eb --- /dev/null +++ b/figs/leg_complexity.pdf diff --git a/figs/re_complexity.pdf b/figs/re_complexity.pdf Binary files differnew file mode 100644 index 0000000..db4ed5a --- /dev/null +++ b/figs/re_complexity.pdf @@ -299,6 +299,20 @@ doi = {10.1103/physrevlett.92.240601} } +@article{Howls_1997_Hyperasymptotics, + author = {Howls, C. J.}, + title = {Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem}, + journal = {Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences}, + publisher = {The Royal Society}, + year = {1997}, + month = {11}, + number = {1966}, + volume = {453}, + pages = {2271--2294}, + url = {https://doi.org/10.1098%2Frspa.1997.0122}, + doi = {10.1098/rspa.1997.0122} +} + @article{Kac_1943_On, author = {Kac, M.}, title = {On the average number of real roots of a random algebraic equation}, @@ -749,7 +749,9 @@ surface is one, and such a surface can divide space into regions. However, in hi After all the work of decomposing an integral into a sub over thimbles, one eventually wants to actually evaluate it. For large $|\beta|$ and in the -absence of any Stokes points, one can come to a nice asymptotic expression. +absence of any Stokes points, one can come to a nice asymptotic expression. For +thorough account of evaluating these integrals (including \emph{at} Stokes +points), see Howls \cite{Howls_1997_Hyperasymptotics}. Suppose that $\sigma\in\Sigma$ is a stationary point at $s_\sigma\in\tilde\Omega$ with a thimble $\mathcal J_\sigma$ that is not involved in any upstream Stokes points. @@ -1388,7 +1390,40 @@ $\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part is zero, then the sign is taken so that the imaginary part of the root has the opposite sign of the imaginary part of $u$. +\cite{Auffinger_2012_Random} +\begin{figure} + \hspace{4pc} + \includegraphics{figs/re_complexity.pdf} + \hspace{-2pc} + \includegraphics{figs/im_complexity.pdf} + \includegraphics{figs/leg_complexity.pdf} + + \caption{ + The complexity of the 3-spin spherical model in the complex plane, as a + function of pure real and imaginary energy (left and right) and the + magnitude $(\operatorname{Im}s)^2/N$ of the distance into the complex + configuration space. The thick black contour shows the line of zero + complexity, where stationary points become exponentially rare in $N$. + } \label{fig:p-spin.complexity} +\end{figure} + +\begin{figure} + \hspace{2pc} + \includegraphics{figs/ground_complexity.pdf} + + \caption{ + The complexity of the 3-spin spherical model in the complex plane, as a + function of pure real energy and the magnitude $(\operatorname{Im}s)^2/N$ + of the distance into the complex configuration space. The thick black + contour shows the line of zero complexity, where stationary points become + exponentially rare in $N$. The shaded region shows where stationary points + have a gapped spectrum. The complexity of the 3-spin model on the real + sphere is shown below the horizontal axis; notice that it does not + correspond with the limiting complexity in the complex configuration space + below the threshold energy. + } \label{fig:ground.complexity} +\end{figure} \subsection{Pure \textit{p}-spin: where are my neighbors?} |