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diff --git a/figs/action.pdf b/figs/action.pdf Binary files differnew file mode 100644 index 0000000..398fc26 --- /dev/null +++ b/figs/action.pdf diff --git a/figs/stationaryPoints.pdf b/figs/stationaryPoints.pdf Binary files differnew file mode 100644 index 0000000..e041c99 --- /dev/null +++ b/figs/stationaryPoints.pdf @@ -147,6 +147,24 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of \emph{all} stationary points of the action, which naturally contains the set of \emph{real} stationary points $\Sigma_0$. +\begin{figure} + \includegraphics{figs/action.pdf}\hfill + \includegraphics{figs/stationaryPoints.pdf} + + \caption{ + An example of a simple action and its critical points. \textbf{Left:} An + action $\mathcal S$ for the $N=2$ spherical (or circular) $3$-spin model, + defined for $s_1,s_2\in\mathbb R$ on the circle $s_1^2+s_2^2=2$ by + $\mathcal S(s_1,s_2)=1.051s_1^3+1.180s_1^2s_2+0.823s_1s_2^2+1.045s_2^3$. In + the example figures in this section, we will mostly use the single angular + variable $\theta=\arctan(s_1,s_2)$, which parameterizes the unit circle and + its complex extension. \textbf{Right:} The stationary points of $\mathcal + S$ in the complex-$\theta$ plane. In this example, + $\Sigma=\{\blacklozenge,\bigstar,\blacktriangle,\blacktriangledown,\bullet,\blacksquare\}$ + and $\Sigma_0=\{\blacklozenge,\blacktriangledown\}$. + } +\end{figure} + Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is orientable, which is usually true) the integral in \eref{eq:partition.function} can be considered an integral over a contour in the complex phase space $\tilde\Omega$, |