Started rereasoning about Stokes connectivity.

3 files changed, 37 insertions, 66 deletions

diff --git a/figs/2_spin_thimbles.pdf b/figs/2_spin_thimbles.pdf
new file mode 100644
index 0000000..f021f98
--- /dev/null
+++ b/figs/2_spin_thimbles.pdf
diff --git a/figs/6_spin.pdf b/figs/6_spin.pdf
new file mode 100644
index 0000000..a04c586
--- /dev/null
+++ b/figs/6_spin.pdf
diff --git a/stokes.tex b/stokes.tex
index 5674fdd..e8c02b6 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -643,72 +643,43 @@ constant complex factor.
 
 \subsection{The conditions for Stokes points}
 
-As we have seen in the previous sections, gradient descent dynamics results in flow that enters and leaves each thimble 
-
-What does the topology of the space of thimbles look like?  Let us consider the
-generic case, where the critical points of $\beta\mathcal S$ have distinct
-energies. Consider initially the situation in the absence of any critical
-point, e.g., as for a constant or linear function. Now, `place' a generic
-(nondegenerate) critical point in the function at $z_0$. In the vicinity of the
-critical point, the flow is locally
-\begin{equation}
-  \dot z
-  \simeq-\frac{\beta^*}2(\operatorname{Hess}\mathcal S|_{z=z_0})^\dagger P(z-z_0)^*
-\end{equation}
-The matrix $(\operatorname{Hess}\mathcal S)^\dagger P$ has a spectrum identical to that of
-$(\operatorname{Hess}\mathcal S)^\dagger$ save marginal directions corresponding to the normals to
-manifold. Assuming we are working in a diagonal basis, this becomes
-\begin{equation}
-  \dot z_i=-\frac12(\beta\lambda_i)^*\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2)
-\end{equation}
-Breaking into real and imaginary parts gives
-\begin{eqnarray}
-  \frac{d\Delta x_i}{dt}&=-\frac12\left(
-    \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i
-  \right) \\
-  \frac{d\Delta y_i}{dt}&=-\frac12\left(
-    \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i
-  \right)
-\end{eqnarray}
-Therefore, in the complex plane defined by each eigenvector of
-$(\operatorname{Hess}\mathcal S)^\dagger P$ there is a separatrix flow.
-
-Continuing to `insert' critical points whose imaginary energy differs from $C$,
-one repeatedly partitions the space this way with each insertion. Therefore,
-for the generic case with $\mathcal N$ critical points, with $C$ differing in
-value from all critical points, the level set $\operatorname{Im}\mathcal S=C$ has
-$\mathcal N+1$ connected components, each of which is simply connected,
-diffeomorphic to $\mathbb R^{2D-1}$ and connects two sectors of infinity to
-each other.
-
-When $C$ is brought to the same value as the imaginary part of some critical
-point, two of these disconnected surfaces pinch grow nearer and pinch together
-at the critical point when $C=\operatorname{Im}\mathcal S$, as in the black lines of
-Figure \ref{fig:local_flow}. The unstable manifold of the critical point, which
-corresponds with the portion of this surface that flows away, produce the
-Lefschetz thimble associated with that critical point.
-
-Stokes lines are trajectories that approach distinct critical points as time
-goes to $\pm\infty$. From the perspective of dynamics, these correspond to
-\emph{heteroclinic orbits}. What are the conditions under which Stokes lines
-appear? Because the dynamics conserves imaginary energy, two critical points
-must have the same imaginary energy if they are to be connected by a Stokes
-line. This is not a generic phenomena, but will happen often as one model
-parameter is continuously varied. When two critical points do have the same
-imaginary energy and $C$ is brought to that value, the level set
-$C=\operatorname{Im}\mathcal S$ sees formally disconnected surfaces pinch together in
-two places. We shall argue that generically, a Stokes line will exist whenever
-the two critical points in question lie on the same connected piece of this
-surface.
-
-What are the ramifications of this for disordered Hamiltonians? When some
-process brings two critical points to the same imaginary energy, whether a
-Stokes line connects them depends on whether the points are separated from each
-other by the separatrices of one or more intervening critical points.
-Therefore, we expect that in regions where critical points with the same
-energies tend to be nearby, Stokes lines will proliferate, while in regions
-where critical points with the same energies tend to be distant compared to
-those with different energies, Stokes lines will be rare.
+As we have seen, gradient descent on the real part of the action results in a
+flow which preserves the imaginary part of the action. Therefore, a necessary
+condition for the existence of a Stokes line between two stationary points is
+for those points to have the same imaginary action. However, this is not a
+sufficient condition. This can be seen in Fig.~\ref{fig:4.spin}, which shows
+the thimbles of the circular 6-spin model. The argument of $\beta$ has been
+chosen such that the stationary points marked by $\clubsuit$ and
+$\blacktriangle$ have exactly the same imaginary energy, and yet they do not
+share a thimble.
+
+\begin{figure}
+  \includegraphics{figs/6_spin.pdf}
+  \caption{
+  Some thimbles of the circular 6-spin model, where the argument of $\beta$ has
+  been chosen such that the imaginary parts of the action at the stationary
+  points $\clubsuit$ and $\blacktriangle$ are exactly the same (and, as a
+  result of conjugation symmetry, the points $\bigstar$ and $\blacksquare$).
+  } \label{fig:4.spin}
+\end{figure}
+
+This is because these stationary points are not adjacent: they are separated
+from each other by the thimbles of other stationary points. This is a
+consistent story in one complex dimension, since the codimension of the
+thimbles is the same as the codimension of the constant imaginary energy
+surface is one, and such a surface can divide space into regions. However, in higher dimensions thimbles do not have codimension high enough to divide space into regions.
+
+\begin{figure}
+  \includegraphics{figs/2_spin_thimbles.pdf}
+  \caption{
+    Thimbles of the $N=3$ spherical 2-spin model projected into the
+    $\operatorname{Re}\theta$, $\operatorname{Re}\phi$,
+    $\operatorname{Im}\theta$ space. The blue and green lines trace gradient
+    descent of the two minima, while the red and orange lines trace those of
+    the two saddles. The location of the maxima are marked as points, but their
+    thimbles are not shown.
+  }
+\end{figure}
 
 \subsection{Evaluating thimble integrals}