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Diffstat (limited to 'stokes.tex')
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@@ -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?} |