From 3c2bde6836189aa8438ca1c05067e6bee82ed324 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 12 Apr 2022 16:05:33 +0200
Subject: Lots of small changes.

---
 stokes.tex | 300 ++++++++++++++++++++++++++++++-------------------------------
 1 file changed, 150 insertions(+), 150 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 8bb6f1a..cc747d9 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -29,8 +29,8 @@
 \begin{abstract}
   In this paper we follow up the study of `complex complex landscapes'
   \cite{Kent-Dobias_2021_Complex}, rugged landscapes of many complex
-  variables.  Unlike real landscapes, there is no useful classification of
-  saddles by index. Instead, the spectrum of fluctuations at stationary points determines their
+  variables.  Unlike real landscapes, the classification of 
+  saddles by index is trivial. 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 suppressed
@@ -40,7 +40,7 @@
   for the analytic continuation of real landscapes with different structures:
   the global minima of `one step replica-symmetry broken' landscapes lie
   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
+  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
   the topological nature of saddles.
@@ -52,36 +52,36 @@
 
 \section{Introduction}
 
-Complex landscapes are basically functions of many variables having many minima
-and, inevitably, many saddles of all `indices' (their number of unstable
-directions). Optimization theory requires us to find the deepest minima, often
+Complex landscapes are functions of many variables having many minima
+and, inevitably, many saddles of all indices (their number of unstable
+directions). Optimization attempts to find the deepest minima, often
 a difficult task.  For example, particles with a repulsive mutual potential
 enclosed in a box will have many stable configurations, and we are asked to
 find the one with lowest energy.
 
-An aim of complexity studies is to be able to classify these landscapes in
+An aim of complexity studies is to classify these landscapes into
 families having common properties. Two simplifications make the task
 potentially tractable. The first is to consider the limit of many variables; in
 the example of the particles, the limit of many particles, i.e. the
 thermodynamic limit.  The
-second simplification is of more technical nature:  we often consider  functions that
-contain some random element to them, and we study the average of an ensemble.
+second simplification is of a more technical nature:  we often consider  functions that
+contain some random element to them, and we study the ensemble average over that randomness.
 The paradigm of this are spin-glasses, where the interactions are random, and
-we are asked to find the ground state energy {\em on average over randomness}.
+we are asked to find the ground state energy for typical samples.
 
-Spin glass theory gave a surprise: random   landscapes come in two kinds:
-those that have a `threshold level' of energy, below which there are many
-minima but almost no saddles, separated by high barriers, and those that have
+Spin glass theory gave a surprise: random landscapes come in two kinds.
+The first kind have a `threshold level' of energy, below which there are many
+minima but almost no saddles, resulting in low minima that are separated by high barriers. The second have
 all sorts of saddles all the way down to the lowest energy levels, and local
 minima are separated by  barriers of sub-extensive energy height.  The latter are still
 complex, but good solutions are easier to find.  This classification is closely
-related to the structure of their Replica Trick solutions.  Armed with this
-solvable random example, it was easy to find non-random examples that behave,
-at least approximately,  in these two ways, for example  sphere packings and the
+related to the structure of their replica trick solutions, the former being `one step replica-symmetry broken' and the latter being `many step replica-symmetry broken.'  Armed with this
+solvable random example, it was easy to find non-random examples that behave
+(at least approximately) in these two ways. For example,  sphere packings and the
 travelling salesman problem belong to first and second classes, respectively.
 
-What about systems whose variables are not real, but rather, complex?  Recalling
-the Cauchy--Riemann conditions, we immediately see a difficulty: if our cost is,
+What about the classification of systems whose variables are not real, but rather, complex?  Recalling
+the Cauchy--Riemann conditions, one finds a difficulty: if our cost is,
 say, the real part of a function of $N$ complex variables, in terms of the
 corresponding $2N$ real variables it has only saddles of index $N$.  Even
 worse: often not all saddles are equally interesting, so simply finding the
@@ -91,33 +91,29 @@ each saddle, there is a `thimble' spanned by the lines along which the cost
 function decreases. The way in which these thimbles fill the complex space is
 crucial for many problems of analytic continuation, and is thus what we need to
 study. The central role played by saddles in a real landscape, the `barriers',
-is now played by the Stokes lines, where thimbles exchange their properties.
+is now played by the Stokes lines, by which thimbles exchange their properties.
 Perhaps not surprisingly, the two classes of real landscapes described above
-retain their role in the complex case, but now  the distinction is that while
+retain their significance in the complex case, but the distinction is now 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,
-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
+and deform it in the many variable complex configuration space.  The landscape one faces
+occupies the entirety of 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 by thimble decomposition}
-\label{sec:thimble.integration}
-
-Analytic continuation of physical theories is sometimes useful. Some theories
+As for analytic continuation of physical theories: it is sometimes useful. Some theories
 have a well-motivated Hamiltonian or action that nevertheless results in a
 divergent partition function, and can only be properly defined by continuation
 from a parameter regime where everything is well-defined
-\cite{Witten_2011_Analytic}. Others result in oscillatory phase space measures
+\cite{Witten_2011_Analytic}. Others result in oscillatory configuration space measures
 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) consists of deforming the phase space integral into a complex phase
+detail) consists of deforming the configuration space integral into a complex configuration
 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
@@ -128,16 +124,20 @@ When your action has a manageable set of stationary points, this process is
 often tractable. However, many actions of interest are complex, having many
 stationary points with no simple symmetry relating them, far too many to
 individually track. Besides appearing in classical descriptions of structural
-and spin glasses, complex landscapes are recently become important objects of
+and spin glasses, complex landscapes have recently become important objects of
 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{Analytic continuation by thimble decomposition}
+\label{sec: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
+Consider an action $\mathcal S$ defined on the (real) configuration space $\Omega$. A
+typical calculation stems from a configuration space average of some observable
 $\mathcal O$ of the form
 \begin{equation} \label{eq:observable}
   \langle\mathcal O\rangle=\frac1Z\int_\Omega ds\,e^{-\beta\mathcal S(s)}\mathcal O(s)
@@ -147,8 +147,7 @@ where the partition function $Z$ normalizes the average as
   Z=\int_\Omega ds\,e^{-\beta\mathcal S(s)}
 \end{equation}
 Rather than focus on any specific observable, we will study the partition
-function itself, since it exhibits the essential features that readily
-generalize to arbitrary observable averages.
+function itself, since it exhibits the essential features.
 
 We've defined $Z$ in a way that suggests application in statistical mechanics,
 but everything here is general: the action can be complex- or even
@@ -157,9 +156,8 @@ contexts, $\Omega$ will be the euclidean real space $\mathbb R^N$ or some
 subspace of this like the sphere $S^{N-1}$ (as in the $p$-spin spherical models
 on which we will treat later). In this paper we will consider only analytic
 continuation of the parameter $\beta$, but any other parameter would work
-equally well, e.g., of some parameter inside the action. The action for real $\beta$ will have,
-along the real direction, 
-some stationary points, e.g., minima, maxima, saddles, and the set of those
+equally well, e.g., of some parameter inside the action. The action for real $\beta$ will have
+some stationary points in the real configuration space, i.e., minima, maxima, saddles, and the set of those
 points in $\Omega$ we will call $\Sigma_0$, the set of real stationary points.
 An example action used throughout this section is shown in
 Fig.~\ref{fig:example.action}.
@@ -173,53 +171,52 @@ Fig.~\ref{fig:example.action}.
   \caption{
     An example of a simple action and its stationary points. \textbf{Left:} The
     configuration space of the $N=2$ spherical (or circular) model, defined for
-    $s\in\mathbb R^N$ restricted to the circle $N=s^2$. It can be parameterized
-    by one angle $\theta=\arctan(s_2/s_1)$. Its natural complex extension takes
-    instead $s\in\mathbb C^N$ restricted to the hyperbola
-    $N=s^2=(\operatorname{Re}s)^2-(\operatorname{Im}s)^2$. The (now complex)
-    angle $\theta$ is still a good parameterization of phase space.
+    $x\in\mathbb R^N$ restricted to the circle $N=x^Tx$. It can be parameterized
+    by one angle $\theta=\arctan(x_2/x_1)$. Its natural complex extension takes
+    instead $z\in\mathbb C^N$ restricted to the hyperbola
+    $N=z^Tz=\|\operatorname{Re}z\|^2-\|\operatorname{Im}z\|^2$. The (now complex)
+    angle $\theta$ is still a good parameterization of configuration space.
     \textbf{Center:} An action $\mathcal S$ for circular $3$-spin model,
     defined by $\mathcal
-    S(s_1,s_2)=-1.051s_1^3-1.180s_1^2s_2-0.823s_1s_2^2-1.045s_2^3$, plotted as
+    S(z_1,z_2)=-1.051z_1^3-1.180z_1^2z_2-0.823z_1z_2^2-1.045z_2^3$, plotted as
     a function of $\theta$. \textbf{Right:} The stationary points of $\mathcal
     S$ in the complex-$\theta$ plane. In this example,
     $\Sigma=\{\mbox{\ding{117}},{\mbox{\ding{72}}},{\mbox{\ding{115}}},{\mbox{\ding{116}}},{\mbox{\ding{108}}},{\mbox{\ding{110}}}\}$
     and $\Sigma_0=\{\mbox{\ding{117}},{\mbox{\ding{116}}}\}$. Symmetries exist
     between the stationary points both as a result of the conjugation symmetry
     of $\mathcal S$, which produces the vertical reflection, and because in the
-    pure 3-spin models $\mathcal S(-s)=-\mathcal S(s)$, which produces the
+    pure 3-spin models $\mathcal S(-z)=-\mathcal S(z)$, which produces the
     simultaneous translation and inversion symmetry.
   } \label{fig:example.action}
 \end{figure}
 
 In order to analytically continue \eref{eq:partition.function}, $\mathcal S$
-must have an extension to a holomorphic function on a larger complex phase
+must have an extension to a holomorphic function on a larger complex configuration
 space $\tilde\Omega$ containing $\Omega$. In many cases this is accomplished by
-simply noticing that the action is some sum or product of holomorphic
+noticing that the action is some sum or product of holomorphic
 functions, e.g., polynomials, and replacing its real arguments with complex
-ones. For $\mathbb R^N$ the complex phase space $\tilde\Omega$ will be $\mathbb
+ones. For $\mathbb R^N$ the complex configuration space $\tilde\Omega$ is $\mathbb
 C^N$, while for the sphere $S^{N-1}$ it takes a little more effort. $S^{N-1}$
 can be defined by all points $x\in\mathbb R^N$ such that $x^Tx=1$. A complex
 extension of the sphere is made by extending this constraint: all points $z\in
-\mathbb C^N$ such that $z^Tz=1$.  Both cases are complex manifolds and moreover
-Kähler manifolds, since they are defined by holomorphic constraints, and
+\mathbb C^N$ such that $z^Tz=1$.  Both cases are complex manifolds, since they are defined by holomorphic constraints, and
 therefore admit a hermitian metric and a symplectic structure. In the extended
-complex phase space, the action often has more stationary points. We'll
+complex configuration space, the action often 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 $\Sigma_0$.
 
-Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is
+Assuming $\mathcal S$ is holomorphic (and that the configuration 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 $\tilde\Omega$,
+can be considered an integral over a contour in the complex configuration space $\tilde\Omega$,
 or
 \begin{equation} \label{eq:contour.partition.function}
   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
+For the moment this translation has only changed a symbol from
+\eref{eq:partition.function}, but conceptually it is important: contour
 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 $\Omega$. This is illustrated in
+contour than our initial configuration space $\Omega$. This is illustrated in
 Fig.~\ref{fig:contour.deformation}.
 
 \begin{figure}
@@ -234,13 +231,13 @@ Fig.~\ref{fig:contour.deformation}.
   \includegraphics{figs/anglepath_3.pdf}
 
   \caption{
-    A schematic picture of the complex phase space for the circular $p$-spin
+    A schematic picture of the complex configuration space for the circular $p$-spin
     model and its standard integration contour. \textbf{Top:} For real variables,
     the model is a circle, and its analytic continuation is a kind of complex
     hyperbola, here shown schematically in three dimensions. \textbf{Bottom:}
     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. \textbf{Left:} The integration contour over the real phase
+    $\theta$ on the circle. \textbf{Left:} The integration contour over the real configuration
     space of the circular model. \textbf{Center:} Complex analysis implies that the
     contour can be freely deformed without changing the value of the integral.
     \textbf{Right:} A funny deformation of the contour in which pieces have been
@@ -249,9 +246,9 @@ Fig.~\ref{fig:contour.deformation}.
   } \label{fig:contour.deformation}
 \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
+What properties are desirable for our contour? Consider the two motivations for performing analytic continuation cited
+in the introduction: we
+want our partition function to be well-defined, i.e., for the configuration space
 integral to converge, and we want to avoid oscillations in the phase of the
 integrand. The first condition, convergence, necessitates that the real part of
 the action $\operatorname{Re}\beta\mathcal S$ be bounded from below, and that it
@@ -261,10 +258,10 @@ $\operatorname{Im}\beta\mathcal S$ be constant.
 
 Remarkably, there is an elegant recipe for accomplishing both these criteria at
 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
+\cite{Witten_2011_Analytic}. We will construct our deformed contour out
 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
+points $\sigma\in\Sigma$ of the action, and it is defined by all points that
 approach the stationary point $s_\sigma$ under gradient descent on
 $\operatorname{Re}\beta\mathcal S$: each thimble is the basin of attraction of a saddle.
 
@@ -277,18 +274,18 @@ integrand as well, a result of the underlying complex geometry of the problem.
 
 What thimbles are necessary to reproduce our original contour, $\Omega$? The
 answer is, we need the minimal set which produces a contour between the same
-places. Simply stated, if $\Omega=\mathbb R$ produced a phase space integral
+places. Simply stated, if $\Omega=\mathbb R$ produced a configuration space integral
 running along the real line from left to right, then our contour must likewise
-take one continuously from left to right, perhaps with detours to well-behaved
+go continuously from left to right, perhaps with detours to well-behaved
 places at infinity (see Fig.~\ref{fig:thimble.homology}). The less simply stated versions follows.
 
 Let $\tilde\Omega_T$ be the set of all points $z\in\tilde\Omega$ such that
-$\operatorname{Re}\beta\mathcal S(z)\geq T$, where we will take $T$ to be a very,
-very large number. $\tilde\Omega_T$ is then the parts of the manifold where it
+$\operatorname{Re}\beta\mathcal S(z)\geq T$, where we will take $T$ to be a
+very large number. $\tilde\Omega_T$ contains the parts of the manifold where it
 is safe for any contour to end up if its integral is to converge, since
 these are the places where the real part of the action is very large and the
 real part of the integrand vanishes exponentially. The relative homology group
-$H_N(\tilde\Omega,\tilde\Omega_T)$ describes the homology of cycles which begin
+$H_N(\tilde\Omega,\tilde\Omega_T)$ describes the homology of $N$-dimensional cycles which begin
 and end in $\Omega_T$, i.e., are well-behaved. Therefore, any well-behaved
 cycle must represent an element of $H_N(\tilde\Omega,\tilde\Omega_T)$. In order
 for our collection of thimbles to produce the correct contour, the composition
@@ -319,7 +316,7 @@ group.
 \end{figure}
 
 Each thimble represents an element of the relative homology, since each thimble
-is a contour on which the real part of the action diverges in any direction.
+is a contour on which the real part of the action diverges at its extremes.
 And, thankfully for us, Morse theory on our complex manifold $\tilde\Omega$
 implies that the set of all thimbles produces a basis for this relative
 homology group, and therefore any contour can be represented by some
@@ -348,7 +345,7 @@ its intersection pairing $n_\sigma=\langle\mathcal C,\mathcal K_\sigma\rangle$.
     direction implied by its orientation, from top to bottom. Therefore,
     $\mathcal C=\mathcal J_{\mbox{\ding{117}}}-\mathcal J_{\mbox{\ding{116}}}$.
     \textbf{Center:} $\arg\beta=0$. Here the thimble of the minimum covers
-    almost all of the real axis, reducing the problem to the real phase space
+    almost all of the real axis, reducing the problem to the real configuration space
     integral. This is also a Stokes point. \textbf{Right:} $\arg\beta=0.1$. Here, one follows the thimble of
     the minimum from left to right again, but now follows that of the maximum
     in the direction implied by its orientation, from bottom to top. Therefore,
@@ -380,7 +377,7 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}.
   \includegraphics{figs/thimble_stokes_3.pdf}
 
   \caption{
-    An example of a Stokes point in the continuation of the phase space
+    An example of a Stokes point in the continuation of the configuration space
     integral involving the action $\mathcal S$ featured in
     Fig.~\ref{fig:example.action}. \textbf{Left:} $\arg\beta=1.176$. The collection of
     thimbles necessary to progress around from left to right, highlighted in a
@@ -393,7 +390,7 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}.
     now $\mathcal J_{\mbox{\ding{115}}}$ must be included. Notice that in this
     figure, because of the symmetry of the pure models, the thimble $\mathcal
     J_{\mbox{\ding{110}}}$ also experiences a Stokes point, but this does not result
-    in a change to the path involving that thimble.
+    in a change to the contour involving that thimble.
   } \label{fig:1d.stokes}
 \end{figure}
 
@@ -410,10 +407,10 @@ keeping track of the resulting weights, and analytic continuation is intractable
 
 The `dynamics' describing thimbles is defined by gradient descent on the real
 part of the action, with a given thimble incorporating all trajectories which
-asymptotically flow to its associated stationary point. Since our phase space
+asymptotically flow to its associated stationary point. Since our configuration space
 is not necessary flat (as for the \emph{spherical} $p$-spin models), we will
-have to do a bit of differential geometry to work out their form. Gradient
-descent on a complex (Kähler) manifold is given by
+have to do a bit of differential geometry to work out the form of the flow. Gradient
+descent on a complex manifold is given by
 \begin{equation} \label{eq:flow.coordinate.free}
   \dot s
   =-\operatorname{grad}\operatorname{Re}\beta\mathcal S
@@ -421,15 +418,15 @@ descent on a complex (Kähler) manifold is given by
   =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s}
 \end{equation}
 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
+$\partial\mathcal S/\partial s^*=0$ because the action is holomorphic. If the complex configuration space is $\mathbb C^N$ and the
 metric is diagonal, this means that the flow is proportional to the conjugate
-of the gradient, or $\dot s\propto-\beta^*(\partial S/\partial s)^*$.
+of the gradient, or $\dot s\propto-\beta^*(\partial\mathcal S/\partial s)^*$.
 
-In the cases we will consider here (namely, that of the spherical models), it
+In the case we will consider here (namely, that of the spherical models), it
 will be more convenient to work in terms of coordinates in a flat embedding
 space than in terms of local coordinates in the curved space, e.g., in terms of
 $z\in\mathbb C^N$ instead of $s\in S^{N-1}$. Let $z:\tilde\Omega\to\mathbb C^N$
-be an embedding of complex phase space into complex euclidean space. The
+be an embedding of complex configuration space into complex euclidean space. The
 dynamics in the embedding space is given by
 \begin{equation}\label{eq:flow.raw}
   \dot z
@@ -479,9 +476,9 @@ to $z$, i.e., $z^\dagger u=0$, $Pu=u$, the identity.
 Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to
 Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$
 and conjugate coordinates given by the real and imaginary parts of each complex
-coordinate. This is because $(\tilde\Omega, g)$ is Kähler and therefore admits
-a symplectic structure, but that the flow conserves
-$\operatorname{Im}\beta\mathcal S$ can be shown using \eref{eq:flow} and the
+coordinate. This is because $(\tilde\Omega, g)$ is a Kähler manifold and therefore admits
+a symplectic structure, but it can be shown that the flow conserves the imaginary action
+using \eref{eq:flow} and the
 holomorphic property of $\mathcal S$:
 \begin{equation}
   \eqalign{
@@ -498,7 +495,8 @@ holomorphic property of $\mathcal S$:
     \right)=0.
   }
 \end{equation}
-A consequence of this conservation is that the flow in the action takes a
+where $\|v\|^2=v^\dagger Pv$ is the norm of a complex vector $v$ in the tangent space of the manifold.
+The flow of the action takes a
 simple form:
 \begin{equation}
   \dot{\mathcal S}
@@ -513,37 +511,29 @@ a direction set by the argument of $\beta$.
 \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
+flow that preserves the imaginary part of the action.
+Stokes lines, when they manifest, are topologically persistent so long as 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
+their connection, i.e., so long as there is not a topological change in the flow. This implies that, despite being relatively low-dimensional
+surfaces of codimension $N$, thimble connections are found with a
 codimension one tuning of parameters, modulo the topological adjacency
-requirement. This means that Stokes points can generically appear when a
+requirement. This means that, though not present in generic cases, Stokes points 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
+Stokes point, but this does not result in a change to the contour involving those
+thimbles. This is because the integer weights 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
@@ -556,11 +546,18 @@ not share a thimble.
   } \label{fig:4.spin}
 \end{figure}
 
+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.
 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
+thimbles is one, and thimbles can divide space into regions. However, in
+higher dimensions thimbles do not have a codimension high enough to divide space
 into regions.
 Nonetheless, thimble intersections
 are still governed by a requirement for adjacency. Fig.~\ref{fig:3d.thimbles}
@@ -572,6 +569,7 @@ 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
@@ -592,10 +590,9 @@ 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
+order for two stationary points to be eligible to share a Stokes point, their thimbles
+must approach the same `good' region of complex configuration space. This is because weight is traded at Stokes points when a facet of
+one thimble flops over another between good regions. 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.
@@ -603,27 +600,27 @@ 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
+Slices of thimbles evaluated at constant `height' as measured by the real part of the
+action are topologically $D-1$ spheres. These slices are known as the \emph{vanishing cycles} of the thimble. At the extremal reaches of the configuration
 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.
+quadratic growth with $D$ of eligible neighbors, something which gives a rough sense of locality in Stokes point interactions.
 
 \subsection{The structure of stationary points}
 \label{subsec:stationary.hessian}
 
-The shape of each thimble in the vicinity of a stationary point can be
+The shape of each thimble in the vicinity of its stationary point can be
 described using an analysis of the hessian of the real part of the action at
 the stationary point. Here we'll review some general properties of this
-hessian, which because the action is holomorphic has rich structure.
+hessian, which has rich structure because the action is holomorphic.
 
 Writing down the hessian using the complex geometry of the previous section
-would be quite arduous. Luckily, we are only interested in the hessian at
-stationary points, and our manifolds of interest are all constraint surfaces.
+would be arduous. Luckily, we are only interested in the hessian at
+stationary points, and our manifolds of interest are constraint surfaces.
 These two facts allow us to find the hessian at stationary points using a
 simpler technique, that of Lagrange multipliers.
 
@@ -642,7 +639,7 @@ $\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\ne
 As a condition for a stationary point, this can be intuited as projecting out
 the normal to the constraint surface $\partial g$ from the gradient of the
 unconstrained action. It implies that the hessian with respect to the complex
-embedding coordinates $z$ at any stationary point is
+embedding coordinate $z$ at any stationary point is
 \begin{equation} \label{eq:complex.hessian}
   \operatorname{Hess}\mathcal S
   =\partial\partial\tilde\mathcal S
@@ -653,11 +650,11 @@ projecting them out using $P$ from the previous section, i.e.,
 $P\operatorname{Hess}\mathcal SP^T$. For notational simplicity we will not
 include this here.
 
-In order to describe the structure of thimbles, one must study the Hessian of
+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
+flow on the real action in the vicinity of stationary points which define them.  We first
 pose the problem as one
-of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$, the hessian of the
+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}
   \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S
@@ -742,15 +739,15 @@ still see it from the second implied equation,
 $(\beta\operatorname{Hess}\mathcal S)^*v^*=\lambda v$, which is the conjugate
 of the first if $\lambda^*=\lambda$.
 
-Something somewhat hidden in the structure of the real hessian but more clear
+Something hidden in the structure of the real hessian but more clear
 in its complex form is that each eigenvalue comes in a pair, since
 \begin{equation}
   \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 vector $iv$, 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
+and ascending directions, i.e., the index of each stationary point is $N$. For
 a stationary point in a real problem this might seem strange, because there are
 clear differences between minima, maxima, and saddles of different index.
 However, for  such a stationary point, its $N$ real Takagi vectors that
@@ -760,7 +757,7 @@ eigenvalue of its partner. A real minimum on the real manifold therefore has
 $N$ downward directions alongside its $N$ upward ones, all pointing directly
 into complex configuration space.
 
-The effect of changing the phase of $\beta$ is revealed by
+The effect of changing the argument of $\beta$ is revealed by
 \eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and
 dividing both sides by $|\beta|e^{i\phi/2}$, one finds
 \begin{equation}
@@ -770,13 +767,13 @@ dividing both sides by $|\beta|e^{i\phi/2}$, one finds
 \end{equation}
 Therefore, one only needs to consider solutions to the Takagi problem for the
 action alone, $\operatorname{Hess}\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the
-resulting Takagi vectors by a constant phase corresponding to half the phase of
-$\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and
+resulting Takagi vectors by a constant phase corresponding to half the argument of
+$\beta$, or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and
 \ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from
-left to right produces a clockwise rotation in the thimbles in the
+left to right produces a clockwise rotation of the thimbles in the
 complex-$\theta$ plane.
 
-These eigenvalues associated with the Takagi vectors can be further related to properties of the
+The eigenvalues associated with the Takagi vectors can be further related to properties of the
 complex symmetric matrix $\beta\operatorname{Hess}\mathcal S$. Suppose that
 $u\in\mathbb R^N$ satisfies the eigenvalue equation
 \begin{equation}
@@ -784,7 +781,7 @@ $u\in\mathbb R^N$ satisfies the eigenvalue equation
   =\sigma u
 \end{equation}
 for some positive real $\sigma$ (because $(\beta\operatorname{Hess}
-S)^\dagger(\beta\operatorname{Hess}\mathcal S)$ is self-adjoint). The square root of these
+S)^\dagger(\beta\operatorname{Hess}\mathcal S)$ is self-adjoint and positive definite). The square root of these
 numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of
 $\beta\operatorname{Hess}\mathcal S$. A direct relationship between these singular
 values and the eigenvalues of the real hessian immediately follows by taking a
@@ -803,13 +800,13 @@ and writing
 Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues
 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.
+eigenvectors of the singular value problem up to a complex factor.
 
 \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
+After all the work of decomposing an integral into a sum over thimbles, one
+eventually wants to evaluate it. For large $|\beta|$ and in the
 absence of any Stokes points, one can come to a nice asymptotic expression. For a
 thorough account of evaluating these integrals (including \emph{at} Stokes
 points), see Howls \cite{Howls_1997_Hyperasymptotics}.
@@ -827,19 +824,20 @@ 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 $u(s):\mathcal J_\sigma\to\mathbb R^D$ such that
+We will make a change of coordinates $u(s):\mathcal J_\sigma\to\mathbb R^D$, where $D$ is the dimension of the manifold ($D=N-1$ for the spherical models),  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}
-\emph{and} the direction of each $\partial u/\partial s$ is along the direction
+\emph{and} the direction of each $\partial u/\partial s$ is aligned with the direction
 of the contour. This is possible because, in the absence of any Stokes points,
 the eigenvectors of the hessian at the stationary point associated with positive
 eigenvalues provide a basis for the thimble. The coordinates $u$ can be real
 because the imaginary part of the action is constant on the thimble, and
 therefore stays with the value it holds at the stationary point, and the real
-part is at its minimum.
+part is at its minimum. The preimage of $u(s)^Tu(s)$ gives the vanishing cycles
+of the thimble, discussed in an earlier subsection.
 
-The coordinates $u$ can be constructed implicitly in the close vicinity of the stationary point, with
+The coordinates $u$ can be constructed implicitly in the close vicinity of the stationary point, with their inverse being
 \begin{equation}
   s(u)=s_\sigma+\sum_{i=1}^{D}\sqrt{\frac{|\beta|}{\lambda^{(i)}}}v^{(i)}u_i+O(u^2)
 \end{equation}
@@ -877,8 +875,9 @@ We therefore have
 \begin{equation}
 Z_\sigma=e^{-\beta\mathcal S(s_\sigma)}\int du\,\det\frac{ds}{du}e^{-\frac{|\beta|}2u^Tu}
 \end{equation}
+which is exact.
 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
+by its value at the stationary point,  and therefore that the determinant can be
 approximated by its value at the stationary point. This gives
 \begin{equation}
   \eqalign{
@@ -892,20 +891,20 @@ We are left with evaluating the determinant of the unitary part of the coordinat
 In circumstances you may be used to, only the absolute value of the determinant
 from the coordinate transformation is relevant, and since the determinant of a
 unitary matrix is always magnitude one, it doesn't enter the computation.
-However, because we are dealing with a path integral, the directions matter,
+However, because we are dealing with a contour integral, the directions matter,
 and there is not an absolute value around the determinant. Therefore, we must
 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, when $\beta=1$ a
+out easily. Take the convention that direction of
+contours along the real line is with the standard orientation. 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
+factor of $e^{-i\phi/2}$ for $\phi=\arg\beta$. Therefore, the contribution for general $\beta$ 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}
@@ -915,7 +914,7 @@ action,
   }
 \end{equation}
 
-We can see that this large-$\beta$ approximation is consistent with the
+We can see that the 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
@@ -933,11 +932,11 @@ 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
+Having introduced the general method for analytic continuation, we will now
+begin dealing with the implications of actions defined in many dimensions
 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
+important to the study of thimbles. Hessians are symmetric matrices by
 construction. For real actions of real variables, the study of random symmetric
 matrices with Gaussian entries provides insight into a wide variety of
 problems. In our case, we will find the relevant ensemble is that of random
@@ -952,7 +951,7 @@ positive, and $|C|\leq\Gamma$. The special case of $C=\Gamma$, where the
 variance of the complex variable and its covariance with its conjugate are the
 same, reduces to the ordinary normal distribution. The case where $C=0$ results
 in the real and imaginary parts of $Z$ being uncorrelated, in what is known as
-the standard complex normal distribution. Its probability density function is
+the standard complex normal distribution. The probability density function for general $\Gamma$ and $C$ is
 defined by
 \begin{equation}
   p(z\mid\Gamma,C)=
@@ -970,7 +969,7 @@ $\overline{XY}=\operatorname{Im}C$.
 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 of the matrices $A$
+constant shift to the  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}
@@ -985,7 +984,7 @@ distributions are shown in the insets of Fig.~\ref{fig:spectra}.
 
 When $C=0$ and the elements of $A$ are standard complex normal, the singular
 value distribution of $B$ is a complex Wishart distribution. For $C\neq0$ the
-problem changes, and to our knowledge a closed form is not in the literature.
+problem changes, and to our knowledge a closed form of the singular value distribution is not in the literature.
 We have worked out an implicit form for the singular value spectrum using the
 replica method, first published in \cite{Kent-Dobias_2021_Complex}.
 
@@ -1012,8 +1011,9 @@ implies a Green function
 \begin{equation}
   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{subsec:stationary.hessian}, using the same linear transformation of
+whose poles give the singular values of $B$.
+This can be put into a manifestly complex form using the method of
+\S\ref{subsec:stationary.hessian}, with the same linear transformation of
 $x,y\in\mathbb R^N$ into $z\in\mathbb C^N$. This gives
 \begin{equation}
   \eqalign{
@@ -1046,7 +1046,7 @@ Green function \cite{Livan_2018_Introduction} gives
     -\sum_\alpha^n\left[z_\alpha^\dagger z_\alpha\sigma
       +\operatorname{Re}\left(z_\alpha^TBz_\alpha\right)
     \right]
-  \right\},
+  \right\}
 \end{equation}
 The average is then made over
 the entries of $B$ and Hubbard--Stratonovich is used to change variables to the
@@ -1075,7 +1075,7 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
   \includegraphics{figs/spectra_15.pdf}
 
   \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 $\overline{A_{ij}^2}=C_0=\frac7{10}e^{i\pi/8}$.
+    Eigenvalue and singular value spectra of a random matrix $B=A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $N\overline{|A_{ij}|^2}=\Gamma_0=1$ and $N\overline{A_{ij}^2}=C_0=\frac7{10}e^{i\pi/8}$.
     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)
@@ -1092,14 +1092,14 @@ are the roots of a sixth-order polynomial, and the root with the smallest value
 of $\operatorname{Re}\alpha_0$ gives the correct solution in all the cases we
 studied. A detailed analysis of the saddle point integration is needed to
 understand why this is so. Evaluated at such a solution, the density of
-singular values follows from the jump across the cut, or
+singular values follows from the jump across the cut in the infinite-$N$ limit, or
 \begin{equation} \label{eq:spectral.density}
   \rho(\sigma)=\frac1{i\pi N}\left(
     \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma)
     -\lim_{\operatorname{Im}\sigma\to0^-}\overline G(\sigma)
   \right)
 \end{equation}
-Examples can be seen in Fig.~\ref{fig:spectra} compared with numeric
+Examples of this distribution can be seen in Fig.~\ref{fig:spectra} compared with numeric
 experiments.
 
 The formation of a gap in the singular value spectrum naturally corresponds to
@@ -1137,7 +1137,7 @@ $p$-tensors whose components are normally distributed with zero mean and
 variance $\overline{J^2}=p!/2N^{p-1}$. The `pure' $p$-spin models have
 $a_i=\delta_{ip}$, 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
+The configuration 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
 manifold by replacing $x$ with $z\in\mathbb C^N$ is holomorphic. The normal to
@@ -2198,7 +2198,7 @@ points become important.
 \label{sec:conclusion}
 
 We have reviewed the Picard--Lefschetz technique for analytically continuing
-integrals and examined its applicability to the analytic continuation of phase
+integrals and examined its applicability to the analytic continuation of configuration
 space integrals over the pure $p$-spin models. The evidence suggests that
 analytic continuation is possible when weight is concentrated in gapped minima,
 who seem to avoid Stokes points, and likely impossible otherwise.
-- 
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