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@@ -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 |