From b4d68122a5f96d02f94043656a1bd8115b529300 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 2 Feb 2022 22:59:59 +0100
Subject: Added figure showing the contour on the complex hyperbola.

---
 stokes.tex | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 6485ed7..37dacbf 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -160,6 +160,30 @@ integrals can have their contour freely deformed (under some constraints)
 without changing their value. This means that we are free to choose a nicer
 contour than our initial phase space $\Omega$.
 
+\begin{figure}
+  \includegraphics{figs/hyperbola_1.pdf}\hfill
+  \includegraphics{figs/hyperbola_2.pdf}\hfill
+  \includegraphics{figs/hyperbola_3.pdf}\\
+  \includegraphics{figs/anglepath_1.pdf}\hfill
+  \includegraphics{figs/anglepath_2.pdf}\hfill
+  \includegraphics{figs/anglepath_3.pdf}
+
+  \caption{
+    A schematic picture of the complex phase space for the circular $p$-spin
+    model and its standard integration contour. (Top, all): For real variables,
+    the model is a circle, and its analytic continuation is a kind of complex
+    hyperbola, here shown schematically in three dimensions. (Bottom, all):
+    Since the real manifold (the circle) is one-dimensional, the complex
+    manifold has one complex dimension, here parameterized by the angle
+    $\theta$ on the circle. (Left): The integration contour over the real phase
+    space of the circular model. (Center): Complex analysis implies that the
+    contour can be freely deformed without changing the value of the integral.
+    (Right): A funny deformation of the contour in which pieces have been
+    pinched off to infinity. So long as no poles have been crossed, even this
+    is legal.
+  }
+\end{figure}
+
 What contour properties are desirable? Consider the two main motivations cited
 in the introduction for performing analytic continuation in the first place: we
 want our partition function to be well-defined, e.g., for the phase space
-- 
cgit v1.2.3-70-g09d2

