Many small edits, and some major writing on Stokes points.

1 files changed, 206 insertions, 151 deletions

diff --git a/stokes.tex b/stokes.tex
index 142b147..8bb6f1a 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -33,8 +33,8 @@
   saddles by index. Instead, the spectrum of fluctuations at stationary points determines their
   topological stability under analytic continuation of the
   theory. Topological changes, which occur at so-called Stokes points, proliferate when the
- saddles have marginal (flat) directions and are exponentially suppressed
-  otherwise. This gives a direct interpretation of the `threshold' energy---which
+ saddles have marginal (flat) directions and are suppressed
+  otherwise. This gives a direct interpretation of the gap or `threshold' energy---which
   in the real case separates saddles from minima--- as the level where the spectrum of
   the Hessian matrix of stationary points develops a gap. This leads to different consequences
   for the analytic continuation of real landscapes with different structures:
@@ -42,7 +42,7 @@
   beyond a threshold, their Hessians are gapped,  and are locally protected from Stokes points, whereas
   those of ``many step replica-symmetry broken'' have gapless Hessians and
   Stokes points immediately proliferate.
-  A new matrix ensemble is found, playing the role that  GUE plays for real landscapes in determining
+  A new matrix ensemble is found, playing the role that GUE plays for real landscapes in determining
   the topological nature of saddles.
 \end{abstract}
 
@@ -97,14 +97,15 @@ retain their role in the complex case, but now  the distinction is that while
 in the first class the Stokes lines among the lowest minima are rare, in the
 second class they proliferate.
 
-In  this paper we shall start from a many-variable integral of a real function,
+In this paper we shall start from a many-variable integral of a real function,
 and deform it in the many variable complex space.  The landscape one faces is
 the full one in this space, and we shall see that this is an example where the
 proliferation -- or lack of it -- of Stokes lines is the interesting quantity
 in this context.
 
 
-\section{Analytic continuation in many variables}
+\section{Analytic continuation by thimble decomposition}
+\label{sec:thimble.integration}
 
 Analytic continuation of physical theories is sometimes useful. Some theories
 have a well-motivated Hamiltonian or action that nevertheless results in a
@@ -115,8 +116,8 @@ that spoil the use of Monte Carlo or saddle point techniques, but can be
 treated in a regime where the measure does not oscillate and the results
 continued to the desired model \cite{Alexandru_2022_Complex}.
 
-In any case, the nicest modern technique (which we will describe in some detail
-later) consists of deforming the phase space integral into a complex phase
+In any case, the nicest modern technique (which we will describe in some
+detail) consists of deforming the phase space integral into a complex phase
 space and then breaking it into pieces associated with stationary points of the
 action. Each of these pieces, known as \emph{thimbles}, has wonderful
 properties that guarantee convergence and prevent oscillations. Once such a
@@ -132,9 +133,8 @@ study in the computer science of machine learning, the condensed matter theory
 of strange metals, and the high energy physics of black holes. What becomes of
 analytic continuation under these conditions?
 
-\section{Thimble integration}
-
 \subsection{Decomposition of the partition function into thimbles}
+\label{subsec:thimble.decomposition}
 
 Consider an action $\mathcal S$ defined on the (real) phase space $\Omega$. A
 typical calculation stems from a phase space average of some observable
@@ -213,7 +213,7 @@ 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 $\tilde\Omega$,
 or
 \begin{equation} \label{eq:contour.partition.function}
-  Z(\beta)=\oint_\Omega ds\,e^{-\beta\mathcal S(s)}
+  Z=\oint_\Omega ds\,e^{-\beta\mathcal S(s)}
 \end{equation}
 For the moment this translation has only changed one of our symbols from
 \eref{eq:partition.function}, but conceptually it is very important: contour
@@ -260,9 +260,9 @@ constant phase, necessitates that the imaginary part of the action
 $\operatorname{Im}\beta\mathcal S$ be constant.
 
 Remarkably, there is an elegant recipe for accomplishing both these criteria at
-once, courtesy of Morse theory. For a more thorough review, see
+once, courtesy of Picard--Lefschetz theory. For a more thorough review, see
 \cite{Witten_2011_Analytic}. We are going to construct our deformed contour out
-of a collection of pieces called \emph{Lefschetz thimbles}, or just thimbles.
+of a collection of pieces called \emph{thimbles}.
 There is one thimble $\mathcal J_\sigma$ associated with each of the stationary
 points $\sigma\in\Sigma$ of the action, and each is defined by all points that
 approach the stationary point $s_\sigma$ under gradient descent on
@@ -359,7 +359,7 @@ its intersection pairing $n_\sigma=\langle\mathcal C,\mathcal K_\sigma\rangle$.
 With these tools in hands, we can finally write the partition function as a sum
 over contributions from each thimble, or
 \begin{equation} \label{eq:thimble.integral}
-  Z(\beta)=\sum_{\sigma\in\Sigma}n_\sigma\oint_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)}.
+  Z=\sum_{\sigma\in\Sigma}n_\sigma\oint_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)}.
 \end{equation}
 Under analytic continuation, the form of \eref{eq:thimble.integral}
 generically persists. When the relative homology of the thimbles is unchanged
@@ -406,6 +406,7 @@ If they occur exponentially often in the system size, there is little hope of
 keeping track of the resulting weights, and analytic continuation is intractable.
 
 \subsection{Gradient flow}
+\label{subsec:thimble.flow}
 
 The `dynamics' describing thimbles is defined by gradient descent on the real
 part of the action, with a given thimble incorporating all trajectories which
@@ -419,8 +420,8 @@ descent on a complex (Kähler) manifold is given by
   =-\left(\frac\partial{\partial s^*}\operatorname{Re}\beta\mathcal S\right)^\sharp
   =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s}
 \end{equation}
-where $g$ is the metric and the holomorphicity of the action was used to set
-$\partial\mathcal S/\partial s^*=0$. If the complex phase space is $\mathbb C^N$ and the
+where $g$ is the metric and
+$\partial\mathcal S/\partial s^*=0$ because the action is holomorphic. If the complex phase space is $\mathbb C^N$ and the
 metric is diagonal, this means that the flow is proportional to the conjugate
 of the gradient, or $\dot s\propto-\beta^*(\partial S/\partial s)^*$.
 
@@ -445,17 +446,11 @@ which is nothing but the projection of $(\partial\mathcal S)^*$ into the
 tangent space of the manifold, with the projection operator
 $P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. Note that $P$ is hermitian.
 
-All one needs to work out the projection operator is a coordinate system. Here we sketch the derivation for the spherical models.
-Take $z_0=e^N$. In a neighborhood of $z_0$, the map $u^\alpha=z^\alpha$ is a coordinate system.
-Its inverse is $z^i=u^i$ for $1\leq i\leq N-1$ and
-\begin{equation}
-  z^N=\sqrt{N-u^2}.
-\end{equation}
-The Jacobian is
-\begin{equation}
-  D_\alpha z^i=\frac{\partial z^i}{\partial u^\alpha}=\delta^i_\alpha-\delta_N^{i}\frac{u^\alpha}{\sqrt{N-u^2}}
-\end{equation}
-Taking the appropriate combination of $Dz$ yields
+Though the projection operator can be derived for any particular manifold by
+defining a coordinate system and computing it with the above definition, for
+simple manifolds like the sphere it can be guessed easily enough, as the unique
+hermitian operator that projects out the direction normal to the surface. For
+the sphere, this is
 \begin{equation}
   P=I-\frac{zz^\dagger}{|z|^2}
 \end{equation}
@@ -514,8 +509,112 @@ simple form:
 In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in
 a direction set by the argument of $\beta$.
 
+\subsection{The conditions for Stokes points}
+\label{subsec:stokes.conditions}
+
+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.
+Stokes lines, when they manifest, are topologically persistent given that this
+conservation is respected: if a Stokes line connects two stationary points and
+the action is smoothly modified under the constraint that the imaginary parts
+of the two stationary points is held equal, the Stokes line will continue to
+connect them so long as the flow of a third stationary point does not sever
+their connection. This implies that despite being relatively low-dimensional
+surfaces of codimension $N$, thimble connections are seen with only a
+codimension one tuning of parameters, modulo the topological adjacency
+requirement. This means that Stokes points can generically appear when a
+dimension-one curve is followed in parameter space.
+
+Not all Stokes points result in the exchange of weight between thimbles.
+Examining Fig.~\ref{fig:1d.stokes} again, notice that the thimbles $\mathcal
+J_{\mbox{\ding{110}}}$ and $\mathcal J_{\mbox{\ding{116}}}$ also experience a
+Stokes point, but this does not result in a change to the path involving those
+thimbles. This is because integer weight can only be modified when a thimble
+that has some nonzero weight is downstream on the gradient descent flow, and
+therefore a necessary condition for a meaningful change in the thimble
+decomposition involving two stationary points $\sigma$ and $\tau$ where
+$n_\sigma\neq0$ and $n_\tau=0$ is for $\operatorname{Re}\beta\mathcal
+S(s_\sigma)<\operatorname{Re}\beta\mathcal S(s_\tau)$.
+
+Another 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
+${\mbox{\ding{115}}}$ have exactly the same imaginary energy, and yet they do
+not share a thimble.
+
+\begin{figure}
+  \hspace{5pc}
+  \hfill
+  \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 ${\mbox{\ding{115}}}$ are exactly the same (and, as a
+  result of conjugation symmetry, the points ${\mbox{\ding{72}}}$ and ${\mbox{\ding{110}}}$).
+  } \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 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.
+Nonetheless, thimble intersections
+are still governed by a requirement for adjacency. Fig.~\ref{fig:3d.thimbles}
+shows a projection of the thimbles of an $N=3$ 2-spin model, which is defined
+on the sphere. Because of an inversion symmetry of the model, stationary points
+on opposite sides of the sphere have identical energies, and therefore also
+share the same imaginary energy. However, their thimbles (blue and green in the
+figure) do not intersect. Here, they could not possibly intersect, since the
+real parts of their energy are also the same, and upward flow could therefore
+not connect them.
+
+\begin{figure}
+  \hspace{5pc}
+  \hfill
+  \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.
+  } \label{fig:3d.thimbles}
+\end{figure}
+
+Determining whether stationary points are adjacent in this sense is a difficult problem,
+known as the global connection problem \cite{Howls_1997_Hyperasymptotics}. It
+is also difficult for us to reason rigorously about the properties of
+stationary point adjacency. However, we have a coarse argument for why, in
+generic cases with random actions, one should expect the typical number of adjacent
+stationary points to scale with a polynomial with dimension. First, notice that in
+order for two stationary points to be eligible to share a Stokes point, they
+must approach the same `good' region of complex configuration space in some
+direction. This is because weight is traded at Stokes points when a facet of
+one thimble flops over another at extremal values. Therefore, one can draw
+conclusions about the number of stationary points eligible for a Stokes point
+with a given stationary point by examining the connectivity of the `good'
+regions.
+
+In the one-dimensional examples above, the `good regions' for contours are
+zero-dimensional, making their topology discrete. However, in a $D$-dimensional
+case, these regions are $D-1$ dimensional, and their topology is richer.
+Thimbles evaluated at constant `height' as measured by the real part of the
+action are topologically $D-1$ spheres. At the extremal reaches of the phase
+space manifold, these spherical slices form a mesh, sharing sections of their
+boundary with the slices of other thimbles and covering the extremal reaches
+like a net. Without some special symmetry to produce vertices in this mesh
+where many thimbles meet, such a mesh generally involves order $D$ boundaries
+coming together in a given place. Considering the number of faces on a given
+extremal slice should also be roughly linear in $D$, one expects something like
+quadratic growth with $D$ of eligible neighbors.
+
 \subsection{The structure of stationary points}
-\label{sec:stationary.hessian}
+\label{subsec:stationary.hessian}
 
 The shape of each thimble in the vicinity of a stationary point can be
 described using an analysis of the hessian of the real part of the action at
@@ -557,7 +656,7 @@ include this here.
 In order to describe the structure of thimbles, one must study the Hessian of
 $\operatorname{Re}\beta\mathcal S$, since it is the upward directions in the
 flow on the real action in the vicinity of stationary points which define the thimbles in the first place.  We first
-pose the problem problem as one
+pose the problem as one
 of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$, the hessian of the
 real part of the action with respect to these real variables is
 \begin{equation} \label{eq:real.hessian}
@@ -649,7 +748,7 @@ in its complex form is that each eigenvalue comes in a pair, since
   \beta\operatorname{Hess}\mathcal S(iv)=i\lambda v^*=-\lambda(iv)
 \end{equation}
 Therefore, if $\lambda$ satisfies \eqref{eq:generalized.eigenproblem} with Takagi vector $v$,
-than so does $-\lambda$, with associated Takagi $iv$, rotated in the complex
+than so does $-\lambda$, with associated Takagi vector $iv$, rotated in the complex
 plane. It follows that each stationary point has an equal number of descending
 and ascending directions, e.g., the index of each stationary point is $N$. For
 a stationary point in a real problem this might seem strange, because there are
@@ -706,74 +805,8 @@ of the real hessian are the singular values of the complex matrix
 $\beta\operatorname{Hess}\mathcal S$, and the Takagi vectors coincide with the
 eigenvectors of the singular value problem up to a constant complex factor.
 
-\subsection{The conditions for Stokes points}
-
-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
-${\mbox{\ding{115}}}$ have exactly the same imaginary energy, and yet they do not
-share a thimble.
-
-\begin{figure}
-  \hspace{5pc}
-  \hfill
-  \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 ${\mbox{\ding{115}}}$ are exactly the same (and, as a
-  result of conjugation symmetry, the points ${\mbox{\ding{72}}}$ and ${\mbox{\ding{110}}}$).
-  } \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}
-  \hspace{5pc}
-  \hfill
-  \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.
-  } \label{fig:3d.thimbles}
-\end{figure}
-
-Despite the fact that in higher dimensions, the level sets of constant
-imaginary energy appear to usually be globally connected, thimble intersections
-are still governed by a requirement for adjacency. Fig.~\ref{fig:3d.thimbles}
-shows a projection of the thimbles of an $N=3$ 2-spin model, which is defined
-on the sphere. Because of an inversion symmetry of the model, stationary points
-on opposite sides of the sphere have identical energies, and therefore also
-share the same imaginary energy. However, their thimbles (blue and green in the
-figure) do not intersect. Here, they could not possibly intersect, since the
-real parts of their energy are also the same, and upward flow could therefore
-not connect them.
-
-Stokes lines, when they manifest, are persistent: if a Stokes line connects two
-stationary points and the action is smoothly modified under the constraint that
-the imaginary parts of the two stationary points is held equal, the Stokes line
-will continue to connect them so long as the flow of a third stationary point
-does not sever their connection. This implies that despite being relatively
-low-dimensional surfaces of codimension $N$, thimble connections are seen with
-only a codimension one tuning of parameters, modulo the topological adjacency
-requirement. This means that Stokes points can generically appear when a
-dimension-one curve is followed in parameter space.
-
 \subsection{Evaluating thimble integrals}
+\label{subsec:thimble.evaluation}
 
 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
@@ -786,7 +819,7 @@ with a thimble $\mathcal J_\sigma$ that is not involved in any upstream Stokes p
 Define its contribution to the partition function (neglecting the integer
 weight) as
 \begin{equation}
-  Z_\sigma(\beta)=\oint_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)}
+  Z_\sigma=\oint_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)}
 \end{equation}
 To evaluate this contour integral in the limit of large $|\beta|$, we will make
 use of the saddle point method, since the integral will be dominated by its
@@ -794,7 +827,7 @@ value at and around the stationary point, where the real part of the action is b
 construction at its minimum on the thimble and the integrand is therefore
 largest.
 
-We will make a change of coordinates $u(s)\in\mathbb R^D$ such that
+We will make a change of coordinates $u(s):\mathcal J_\sigma\to\mathbb R^D$ such that
 \begin{equation} \label{eq:thimble.integration.def}
   \beta\mathcal S(s)=\beta\mathcal S(s_\sigma)+\frac{|\beta|}2 u(s)^Tu(s)
 \end{equation}
@@ -830,28 +863,29 @@ to confirm that these coordinates satisfy \eqref{eq:thimble.integration.def} asy
 \end{equation}
 The Jacobian of this transformation is
 \begin{equation}
-  \frac{\partial s_i}{\partial u_j}=\sqrt{\frac{|\beta|}{\lambda^{(j)}}}v^{(j)}_i
-  =\sqrt{\frac1{\lambda_0^{(j)}}}v^{(j)}_i
+  \frac{\partial s_i}{\partial u_j}=\sqrt{\frac{|\beta|}{\lambda^{(j)}}}v^{(j)}_i+\cdots
+  =\sqrt{\frac1{\lambda_0^{(j)}}}v^{(j)}_i+\cdots
 \end{equation}
-where $\lambda_0=\lambda/|\beta|$ is the eigenvalue of the hessian for
-$\beta=1$. This is a $N\times D$ matrix. Since the eigenvectors $v$ are
-mutually complex orthogonal, $v_i^{(j)}$ is nearly a unitary matrix, and it can
-be made one by including $v^{(N)}=\widehat{\partial g}$, the unit normal to the
-constraint manifold. This lets us write $U_{ij}=v_i^{(j)}$ an $N\times N$
+where $\lambda_0^{(j)}=\lambda^{(j)}/|\beta|$ is the $j$th eigenvalue of the
+hessian evaluated at the stationary point for $\beta=1$. This is na\"ively an
+$N\times D$ matrix, because the Takagi vectors are $N$ dimensional, but care
+must be taken to project each into the tangent space of the manifold to produce
+a $D\times D$ matrix. This lets us write $U_{ij}=v_i^{(j)}$ a $D\times D$
 unitary matrix, whose determinant will give the correct phase for the measure.
 
 We therefore have
 \begin{equation}
-Z_\sigma(\beta)=e^{-\beta\mathcal S(s_\sigma)}\int du\,\det\frac{ds}{du}e^{-\frac{|\beta|}2u^Tu}
+Z_\sigma=e^{-\beta\mathcal S(s_\sigma)}\int du\,\det\frac{ds}{du}e^{-\frac{|\beta|}2u^Tu}
 \end{equation}
 Now we take the saddle point approximation, assuming the integral is dominated
 by its value at the stationary point such that the determinant can be
 approximated by its value at the stationary point. This gives
 \begin{equation}
   \eqalign{
-    Z_\sigma(\beta)
+    Z_\sigma
       &\simeq e^{-\beta\mathcal S(s_\sigma)}\left.\det\frac{ds}{du}\right|_{s=s_\sigma}\int du\,e^{-\frac{|\beta|}2u^Tu} \\
-      &=e^{-\beta\mathcal S(s_\sigma)}\left(\prod_i^D\sqrt{\frac1{\lambda_0^{(i)}}}\right)\det U\left(\frac{2\pi}{|\beta|}\right)^{D/2}
+      &=e^{-\beta\mathcal S(s_\sigma)}\left(\prod_i^D\sqrt{\frac1{\lambda_0^{(i)}}}\right)\det U\left(\frac{2\pi}{|\beta|}\right)^{D/2} \\
+      &=e^{-\beta\mathcal S(s_\sigma)}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-1/2}\det U\left(\frac{2\pi}{|\beta|}\right)^{D/2}
   }
 \end{equation}
 We are left with evaluating the determinant of the unitary part of the coordinate transformation.
@@ -864,28 +898,44 @@ determine the phase that it contributes.
 
 This is difficult in general, but for real stationary points it can be reasoned
 out easily. Take the same convention we used earlier, that the direction of
-contours along the real line is in the conventional directions. Then, a
-stationary point of index $k$ has $k$ real eigenvectors and $D-k$ purely
-imaginary eigenvectors that contribute to its thimble. The matrix of
-eigenvectors can therefore be written $U=i^kO$ for an orthogonal matrix $O$,
-and with all eigenvectors canonically oriented $\det O=1$. We therefore have
-$\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action before any Stokes points,
+contours along the real line is in the conventional directions. Then, when $\beta=1$ a
+stationary point of index $k$ has $D-k$ real Takagi vectors and $k$ purely
+imaginary Takagi vectors that correspond with upward directions in the flow
+and contribute to its thimble. The matrix of Takagi vectors can therefore be
+written $U=i^kO$ for an orthogonal matrix $O$, and with all eigenvectors
+canonically oriented $\det O=1$. We therefore have $\det U=i^k$ when $\beta=1$. As the
+argument of $\beta$ is changed, we know how the eigenvectors change: by a
+factor of $e^{-i\phi/2}$ for $\phi=\arg\beta$. Therefore, the contribution more generally is
+$\det U=(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real
+action,
 \begin{equation} \label{eq:real.thimble.partition.function}
-  Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)}
+  \eqalign{
+    Z_\sigma&\simeq\left(\frac{2\pi}{|\beta|}\right)^{D/2}e^{-i\phi D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} \\
+                   &=\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} \\
+  }
 \end{equation}
 
+We can see that this large-$\beta$ approximation is consistent with the
+relationship between thimble orientation and integer weight outlined in
+Fig.~\ref{fig:thimble.orientation}. There, it is seen that taking the argument
+of $\beta$ through zero results in a series of Stokes points among real
+stationary points of a real action which switches the sign of the integer
+weights of thimbles with odd index and preserves the integer weights of
+thimbles with even index. For an real action, taking $\beta\to\beta^*$ should
+simply take $Z\to Z^*$. Using the formula above, we find
 \begin{equation}
   Z(\beta)^*
   =\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta)^*
   =\sum_{\sigma\in\Sigma_0}n_\sigma(-1)^{k_\sigma}Z_\sigma(\beta^*)
   =Z(\beta^*)
 \end{equation}
+as expected.
 
 \section{The ensemble of symmetric complex-normal matrices}
 
 Having introduced the generic method for analytic continuation, we will now
 begin dealing with the implications of actions defined in very many dimensions
-with disorder.  We saw in \S\ref{sec:stationary.hessian} that the singular
+with disorder.  We saw in \S\ref{subsec:stationary.hessian} that the singular
 values of the complex hessian of the action at each stationary point are
 important in the study of thimbles. Hessians are symmetric matrices by
 construction. For real actions of real variables, the study of random symmetric
@@ -920,7 +970,7 @@ $\overline{XY}=\operatorname{Im}C$.
 We will consider an ensemble of random $N\times N$ matrices $B=A+\lambda_0I$, where the
 entries of $A$ are complex-normal distributed with variances
 $\overline{|A_{ij}|^2}=\Gamma_0/N$ and $\overline{A_{ij}^2}=C_0/N$, and $\lambda_0$ is a
-constant shift to its diagonal. The eigenvalue distribution the matrices $A$
+constant shift to its diagonal. The eigenvalue distribution of the matrices $A$
 is already known to take the form of an elliptical ensemble in the large-$N$
 limit, with constant support inside the ellipse defined by
 \begin{equation} \label{eq:ellipse}
@@ -945,7 +995,7 @@ eigenvalues of the real $2N\times2N$ block matrix
 \begin{equation}
   \left[\matrix{\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right]
 \end{equation}
-as we saw in \S\ref{sec:stationary.hessian}. The $2N\times2N$ problem is easier
+as we saw in \S\ref{subsec:stationary.hessian}. The $2N\times2N$ problem is easier
 to treat analytically than the $N\times N$ one because the matrix under study
 is linear in the entries of $B$. The eigenvalue spectrum of this block matrix
 can be studied by ordinary techniques from random matrix theory.  Defining the `partition function'
@@ -963,7 +1013,7 @@ implies a Green function
   G(\sigma)=\frac\partial{\partial\sigma}\log Z(\sigma)
 \end{equation}
 This can be put into a manifestly complex form in the same way it was done in
-\S\ref{sec:stationary.hessian}, using the same linear transformation of
+\S\ref{subsec:stationary.hessian}, using the same linear transformation of
 $x,y\in\mathbb R^N$ into $z\in\mathbb C^N$. This gives
 \begin{equation}
   \eqalign{
@@ -988,7 +1038,7 @@ $x,y\in\mathbb R^N$ into $z\in\mathbb C^N$. This gives
 which is a general expression for the singular values $\sigma$ of a symmetric
 complex matrix $B$.
 
-Introducing replicas to bring the partition function into the numerator of the
+Introducing replicas to eliminate the logarithm in the
 Green function \cite{Livan_2018_Introduction} gives
 \begin{equation} \label{eq:green.replicas} \fl\quad
   G(\sigma)=\lim_{n\to0}\int dz^*dz\,z_0^\dagger z_0
@@ -1026,7 +1076,7 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
 
   \caption{
     Eigenvalue and singular value spectra of a random matrix $B=A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $\overline{|A_{ij}|^2}=\Gamma_0=1$ and $\overline{A_{ij}^2}=C_0=\frac7{10}e^{i\pi/8}$.
-    The diaginal shifts differ in each plot, with (a) $\lambda_0=0$, (b)
+    The diagonal shifts differ in each plot, with (a) $\lambda_0=0$, (b)
     $\lambda_0=\frac12|\lambda_{\mathrm{gap}}|$, (c)
     $\lambda_0=|\lambda_{\mathrm{gap}}|$, and (d)
     $\lambda_0=\frac32|\lambda_{\mathrm{gap}}|$. The shaded region of each
@@ -1071,6 +1121,7 @@ depends on the argument of $\lambda_0$, or the direction in the complex plane
 in which the distribution is shifted.
 
 \section{The \textit{p}-spin spherical models}
+\label{sec:p-spin}
 
 The $p$-spin spherical models are defined by the action
 \begin{equation} \label{eq:p-spin.hamiltonian}
@@ -1088,7 +1139,7 @@ $a_i=\delta_{ip}$, while the mixed have some more complicated coefficients $a$.
 
 The phase space manifold $\Omega=\{x\mid x^2=N, x\in\mathbb R^N\}$ has a
 complex extension $\tilde\Omega=\{z\mid z^2=N, z\in\mathbb C^N\}$. The natural
-extension of the hamiltonian \eref{eq:p-spin.hamiltonian} to this complex
+extension of the Hamiltonian \eref{eq:p-spin.hamiltonian} to this complex
 manifold by replacing $x$ with $z\in\mathbb C^N$ is holomorphic. The normal to
 this manifold at any point $z\in\tilde\Omega$ is always in the direction $z$.
 The projection operator onto the tangent space of this manifold is given by
@@ -1111,6 +1162,7 @@ which for the pure $p$-spin in particular implies that $\mu=p\epsilon$ for
 specific energy $\epsilon$.
 
 \subsection{2-spin}
+\label{subsec:2-spin}
 
 The pure 2-spin model is diagonalizable and therefore exactly solvable, and is
 not complex in the sense of having a superextensive number of stationary points
@@ -1171,13 +1223,8 @@ points in the 2-spin model with the same imaginary energy will possess one.
 These trajectories are plotted in Fig.~\ref{fig:two-spin}.
 
 \begin{figure}
-  \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
-    set xlabel '$\Delta t$'
-    set ylabel '$z(t) / \sqrt{N}$'
+  \hfill\includegraphics{figs/two_spin_thimbles.pdf}
 
-    plot 1 / sqrt(1 + exp(2 * x)) t '$z_1$', \
-         1 / sqrt(1 + exp(- 2 * x)) t '$z_2$'
-  \end{gnuplot}
   \caption{
     The Stokes line in the 2-spin model when the stationary points associated
     with the first and second cardinal directions are brought to the same
@@ -1209,7 +1256,7 @@ partition function contribution from the thimble of a real stationary point, we
 have
 \begin{equation}
   \eqalign{
-    Z(\beta)
+    Z
     &=\int_{S^{N-1}}ds\,e^{-\beta\mathcal S_2(s)}
     =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S_2(s)} \\
     &\simeq\sum_{\sigma\in\Sigma_0}i^{k_\sigma}\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta\mathcal S_2(s_\sigma)}|\det\operatorname{Hess}\mathcal S_2(s_\sigma)|^{-\frac12} \\
@@ -1221,7 +1268,7 @@ have
 where $\epsilon_k$ is the energy of the twin stationary points of index $k$. In the large $N$ limit, we take advantage of the limiting distribution $\rho$ of these energies to write
 \begin{equation} \fl
   \eqalign{
-    \overline Z(\beta)
+    \overline Z
     &=2\int d\epsilon\,\rho(\epsilon)\exp\left\{
       i\frac\pi2k_\epsilon+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon-\frac D2\int d\epsilon'\,\rho(\epsilon')\log2|\epsilon-\epsilon'|
     \right\} \\
@@ -1295,6 +1342,7 @@ sections, complicated by the additional presence of Stokes points in the
 continuation.
 
 \subsection{Pure \textit{p}-spin: where are the saddles?}
+\label{subsec:p-spin.one.replica}
 
 We studied the distribution of stationary points in the pure $p$-spin models in
 a previous work \cite{Kent-Dobias_2021_Complex}. Here, we will review the
@@ -1326,7 +1374,7 @@ more generic form in the complex case, where now the threshold separates
 stationary points that have mostly gapped from mostly ungapped spectra. Since
 the $p$-spin model has a Hessian that consists of a symmetric complex matrix
 with a shifted diagonal, we can use the results of
-\S\ref{sec:stationary.hessian}. The variance of the $p$-spin hessian without
+\S\ref{subsec:stationary.hessian}. The variance of the $p$-spin hessian without
 shift is
 \begin{equation}
   \overline{|\partial\partial\mathcal S_p|^2}
@@ -1341,7 +1389,7 @@ shift is
 where $Y=\frac1N\|\operatorname{Im}z\|^2$ is a measure of how far the
 stationary point is into the complex configuration space.  As expected for a
 real problem, the two variances coincide when $Y=0$.  The diagonal shift is
-$-p\epsilon$. In the language of \S\ref{sec:stationary.hessian}, this means
+$-p\epsilon$. In the language of \S\ref{subsec:stationary.hessian}, this means
 that $\Gamma_0=p(p-1)(1+2Y)^{p-2}/2$, $C_0=p(p-1)/2$, and
 $\lambda_0=-p\epsilon$. This means that the energy at which the gap appears is,
 using \eqref{eq:gap.eigenvalue},
@@ -1369,7 +1417,7 @@ trick. Based on the experience from similar problems
 \sim \log \overline{ \mathcal N}$ is expected to be exact wherever the
 complexity is positive.
 
-As in \S\ref{sec:stationary.hessian}, this expression can be bright into a
+As in \S\ref{subsec:stationary.hessian}, this expression can be bright into a
 manifestly complex form using Cauchy--Riemann relations. This gives
 \begin{equation}
   \mathcal N
@@ -1386,7 +1434,7 @@ superfields applied to the $p$-spin spherical models, see
 does not use superfields \cite{Kent-Dobias_2021_Complex}, but they will be
 essential for compactly writing the \emph{two} replica complexity in the next
 section, and so we briefly introduce the technique here. Introducing the
-one-component Grassman variables $\theta$ and $\bar\theta$, define the
+one-component Grassmann variables $\theta$ and $\bar\theta$, define the
 superfield
 \begin{equation}
   \phi(1)=z+\bar\theta(1)\eta+\gamma\theta(1)+\hat z\bar\theta(1)\theta(1)
@@ -1399,9 +1447,9 @@ Then the expression for the number of stationary points can be written in a very
       \tilde\mathcal S_p(\phi(1))
   \right\}
 \end{equation}
-where $d1=d\bar\theta(1)\,d\theta(1)$ denotes the integration over the grasssman variables.
+where $d1=d\bar\theta(1)\,d\theta(1)$ denotes the integration over the Grassmann variables.
 This can be related to the previous expression by expansion with respect to
-the Grassman variables, recognizing that $\theta^2=\bar\theta^2=0$ restricts
+the Grassmann variables, recognizing that $\theta^2=\bar\theta^2=0$ restricts
 the series to two derivatives.
 
 From here the process can be treated as usual, averaging over the couplings and
@@ -1575,6 +1623,7 @@ are the only stationary points that are somewhat protected from participation
 in Stokes points.
 
 \subsection{Pure \textit{p}-spin: where are my neighbors?}
+\label{subsec:p-spin.two.replica}
 
 The problem of counting the density of Stokes points in an analytic
 continuation of the spherical models is quite challenging, as the problem of
@@ -1698,7 +1747,7 @@ Define $\theta_{xx}$ as the angle between $x_1$ and $x_2$. Then $x_1^Tx_2=\|x_1\
 \begin{equation}
   \Delta=2(1+\|y_2\|^2-\sqrt{1-\|y_2\|^2}\cos\theta_{xx})
 \end{equation}
-The definite of $\gamma$ likewise gives
+The definition of $\gamma$ likewise gives
 \begin{equation}
   \eqalign{
     \gamma\Delta&=2-2x_1^Tx_2-2ix_1^Ty_2=2(1-|x_2|\cos\theta_{xx}-i|y_2|\cos\theta_{xy}) \\
@@ -1843,7 +1892,7 @@ peak is at $90^\circ$ but has a height equal to $\Sigma_{k=1}$, the complexity o
 rank-1 saddles.
 
 \begin{figure}
-  \includegraphics{figs/neighbor_energy_limit.pdf}
+  \hfill\includegraphics{figs/neighbor_energy_limit.pdf}
   \caption{
     The two-replica complexity $\Sigma^{(2)}$ scaled by $\Sigma_{k=1}$ as a function of
     angle $\varphi$ for various $\Delta$ at $\epsilon_1=\epsilon_{k=2}$, the
@@ -1868,7 +1917,10 @@ stationary point gives some hope for the success of continuation involving
 these points: since Stokes points only lead to a change in weight when they
 involve upward flow from a point that already has weight, neighbors that have a
 lower energy won't be eligible to be involved in a Stokes line that causes a
-change of weight until the phase of $\beta$ has rotated almost $180^\circ$.
+change of weight until the phase of $\beta$ has rotated almost $180^\circ$. The
+energy of nearest neighbors is plotted in Fig.~\ref{fig:neighbor.energy}, while
+their angular distribution and distance is plotted in
+Fig.~\ref{fig:nearest.properties}.
 
 \begin{figure}
   \includegraphics{figs/neighbor_closest_energy.pdf}
@@ -1879,11 +1931,11 @@ change of weight until the phase of $\beta$ has rotated almost $180^\circ$.
     dashed line shows $\epsilon_2=\epsilon_1$. The nearest neighbor energy
     coincides with the dashed line until $\epsilon_{k=1}$, the energy where
     rank-one saddles vanish, where it peels off.
-  }
+  } \label{fig:neighbor.energy}
 \end{figure}
 
 \begin{figure}
-  \includegraphics{figs/neighbor_plot.pdf}
+  \hfill\includegraphics{figs/neighbor_plot.pdf}
 
   \caption{
     The properties of the nearest neighbor saddles in the 3-spin model as a function of energy
@@ -1895,11 +1947,12 @@ change of weight until the phase of $\beta$ has rotated almost $180^\circ$.
     points are found at arbitrarily close distance but only at $90^\circ$.
     Below $\epsilon_{k=1}$, neighboring stationary points are separated by a
     minimum squared distance $\Delta_\textrm{min}$, and the angle they are
-    found at drifts.
+    found at drifts. The complexity of nearest neighbors in the shaded region is $\Sigma_{k\geq2}$, while in along the solid line for $\epsilon>\epsilon_{k=1}$ it is $\Sigma_{k=1}$. Below $\epsilon_{k=1}$ the complexity of nearest neighbors is zero.
   } \label{fig:nearest.properties}
 \end{figure}
 
 \subsection{Pure {\it p}-spin: numerics}
+\label{subsec:p-spin.numerics}
 
 To study Stokes lines numerically, we approximated them by parametric curves.
 If $z_0$ and $z_1$ are two stationary points of the action with
@@ -1967,7 +2020,7 @@ spectrum of the stationary point in question on the likelihood that a randomly
 chosen neighbor will share a Stokes line.
 
 \begin{figure}
-  \includegraphics{figs/numerics_prob_eigenvalue.pdf}
+  \hfill\includegraphics{figs/numerics_prob_eigenvalue.pdf}
 
   \caption{
     The probability $P_\mathrm{Stokes}$ that a real stationary point will share
@@ -1989,7 +2042,7 @@ everywhere expect where $\lambda_\textrm{min}\ll1$. This supports the idea that
 gapped minima are unlikely to see Stokes lines.
 
 \begin{figure}
-  \includegraphics{figs/numerics_angle_gap_32.pdf}
+  \hfill\includegraphics{figs/numerics_angle_gap_32.pdf}
 
   \caption{
     The probability density function for identified Stokes points as a function
@@ -2013,6 +2066,7 @@ concentrated at phases that are nearly $180^\circ$, where the two-replica
 calculation shows that almost all of their nearest neighbors will lie.
 
 \subsection{Pure {\it p}-spin: is analytic continuation possible?}
+\label{subsec:p-spin.continuation}
 
 After this work, one is motivated to ask: can analytic continuation be done in
 even a simple complex model like the pure $p$-spin? Numeric and analytic
@@ -2028,12 +2082,12 @@ weight is concentrated in precisely these points.
 Recalling our expression of the single-thimble contribution to the partition
 function for a real stationary point of a real action expanded to lowest order
 in large $|\beta|$ \eqref{eq:real.thimble.partition.function}, we can write for
-the $p$-spin after an infintesimal rotation of $\beta$ into the complex plane
+the $p$-spin after an infinitesimal rotation of $\beta$ into the complex plane
 (before any Stokes points have been encountered)
 \begin{equation}
   \eqalign{
-    Z(\beta)
-    &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta) \\
+    Z
+    &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma \\
     &\simeq\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}
       |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}
       e^{-\beta\mathcal S(s_\sigma)} \\
@@ -2115,7 +2169,7 @@ the transition for finite $|\beta|$, which likely results from the invalidity
 of our large-$\beta$ approximation. More of the phase diagram might be
 constructed by continuing the series for individual thimbles to higher powers
 in $\beta$, which would be equivalent to allowing non-constant terms in the
-Jacobian over the thimble.
+Jacobian of the coordinate transformation over the thimble.
 
 \begin{figure}
   \hspace{5pc}
@@ -2141,6 +2195,7 @@ thimbles to the next order in $\beta$ may reveal more explicitly where Stokes
 points become important.
 
 \section{Conclusion}
+\label{sec:conclusion}
 
 We have reviewed the Picard--Lefschetz technique for analytically continuing
 integrals and examined its applicability to the analytic continuation of phase
@@ -2159,11 +2214,11 @@ It is possible that a statistical theory of analytic continuation could be
 developed in order to treat these cases, whereby one computes the average or
 typical rate of Stokes points as a function of stationary point properties, and
 treats their proliferation to complex saddles as a structured diffusion
-problem. This would be a very involved calculation, involving counting exact
+problem. This would be a very involved calculation, involving counting
 classical trajectories with certain boundary conditions, but in principle it
 could be done as in \cite{Ros_2021_Dynamical}. Here the scale of the
-proliferation may save things to a degree, allowing accurate statements to be
-made about its average effects.
+proliferation may rescue things, allowing accurate statements to be
+made about its average effect.
 
 \section*{References}
 \bibliographystyle{unsrt}