From a22eaec23a565a8906b65aefa0f37687be2f2dd4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Mar 2022 16:41:15 +0100
Subject: Updated first sections of paper.

---
 stokes.tex | 280 +++++++++++++++++++++++++++++++++----------------------------
 1 file changed, 154 insertions(+), 126 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index ba5a11e..0973242 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -11,6 +11,7 @@
 ]{hyperref} % ref and cite links with pretty colors
 \usepackage{amsopn, amssymb, graphicx, xcolor} % standard packages
 \usepackage[subfolder]{gnuplottex} % need to compile separately for APS
+\usepackage{pifont}
 
 \begin{document}
 
@@ -51,46 +52,56 @@
 
 \section{Preamble}
 
-Complex landscapes are basically functions of many variables having many minima and, inevitably,
-many saddles of all `indices' (their number of unstable directions). Optimization problems 
-require us 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 theory is to be able to classify these landscapes in 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)
-may be expected to bring about simplifications.
-The second simplification is of more technical nature:  we consider  functions that contain some random element to them, and we
-study the average of an ensemble. The paradigm of this is the spin-glass, where the interactions are random, and we are asked to find the
-ground state energy {\em on average over randomness}.
-
-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 all sorts of saddles all the way down to the lowest 
-energy levels, and local minima are separated by relatively small barriers.
-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 (e.g. 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, 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 lowest is not usually what we
-need to do (more about this below).
-As it turns out, there is a set of interesting questions to ask, as we describe below. For 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. 
-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 in the first class the Stokes lines 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 proliferation -- or lack of it -- 
-of Stokes lines is the interesting quantity  in this context.
+Complex landscapes are basically functions of many variables having many minima
+and, inevitably, many saddles of all `indices' (their number of unstable
+directions). Optimization problems require us 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 theory is to be able to classify these landscapes in
+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) may be expected to bring about simplifications.  The
+second simplification is of more technical nature:  we consider  functions that
+contain some random element to them, and we study the average of an ensemble.
+The paradigm of this is the spin-glass, where the interactions are random, and
+we are asked to find the ground state energy {\em on average over randomness}.
+
+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
+all sorts of saddles all the way down to the lowest energy levels, and local
+minima are separated by relatively small barriers.  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 (e.g. 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,
+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
+lowest is not usually what we need to do (more about this below).  As it turns
+out, there is a set of interesting questions to ask, as we describe below. For
+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.
+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
+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
+proliferation -- or lack of it -- of Stokes lines is the interesting quantity
+in this context.
 
 
 \section{Introduction}
@@ -98,10 +109,11 @@ of Stokes lines is the interesting quantity  in this context.
 Analytic continuation of physical theories 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{}. Others result
-in oscillatory phase 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
-oscillated and the results continued to the desired model \cite{Alexandru_2022_Complex}.
+from a parameter regime where everything is well-defined
+\cite{Witten_2011_Analytic}. Others result in oscillatory phase 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
 later) consists of deforming the phase space integral into a complex phase
@@ -114,7 +126,7 @@ where the thimbles interact, which must be accounted for.
 When your action has a manageable set of stationary points, this process is
 usually tractable. However, many actions of interest are complex, having many
 stationary points with no simple symmetry relating them, far too many to
-individually monitor. Besides appearing in classical descriptions of structural
+individually track. Besides appearing in classical descriptions of structural
 and spin glasses, complex landscapes are 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
@@ -125,55 +137,53 @@ analytic continuation under these conditions?
 \subsection{Decomposition of the partition function into thimbles}
 
 Consider an action $\mathcal S$ defined on the (real) phase space $\Omega$. A
-typical calculation stems from the partition function
+typical calculation stems from a phase 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)
+\end{equation}
+where the partition function $Z$ normalizes the average as
 \begin{equation} \label{eq:partition.function}
-  Z(\beta)=\int_\Omega ds\,e^{-\beta\mathcal S(s)}
+  Z=\int_\Omega ds\,e^{-\beta\mathcal S(s)}
 \end{equation}
-We've defined $Z$ in a way that strongly suggests application in statistical
-mechanics, but everything here is general: the action can be complex- or even
+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.
+
+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
 imaginary-valued, and $\Omega$ could be infinite-dimensional. In typical
 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
-we will study later). We will consider only the analytic continuation of the
-parameter $\beta$, but any other would work equally well, e.g., of some
-parameter inside the action. The action will have some stationary points, e.g.,
-minima, maxima, saddles, and the set of those points in $\Omega$ we will call
-$\Sigma_0$, the set of real stationary points.
-
-In order to analytically continue \eref{eq:partition.function} by the method we
-will describe, $\mathcal S$ must have an extension to a holomorphic function on
-a larger complex phase space $\tilde\Omega$ containing $\Omega$. In many cases
-this is accomplished by simply noticing that the action is some sum or product
-of holomorphic functions, e.g., polynomials. For $\mathbb R^N$ the complex
-phase space $\tilde\Omega$ will be $\mathbb C^N$, 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 therefore admit a hermitian
-metric and a symplectic structure. In the extended complex phase space, the
-action potentially 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 $\Sigma_0$.
+on which we will focus). 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 will have
+some stationary points, e.g., 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}.
 
 \begin{figure}
   \hspace{5pc}
+  \includegraphics{figs/circle.pdf}\hfill
   \includegraphics{figs/action.pdf}\hfill
   \includegraphics{figs/stationaryPoints.pdf}
 
   \caption{
-    An example of a simple action and its stationary points. \textbf{Left:} An
-    action $\mathcal S$ for the $N=2$ spherical (or circular) $3$-spin model,
-    defined for $s_1,s_2\in\mathbb R$ on the circle $s_1^2+s_2^2=2$ by
-    $\mathcal S(s_1,s_2)=-1.051s_1^3-1.180s_1^2s_2-0.823s_1s_2^2-1.045s_2^3$. In
-    the example figures in this section, we will mostly use the single angular
-    variable $\theta$ defined by $s_1=\sqrt2\cos\theta$,
-    $s_2=\sqrt2\sin\theta$, which parameterizes the unit circle and its complex
-    extension, as $\cos^2\theta+\sin^2\theta=1$ is true even for complex
-    $\theta$. \textbf{Right:} The stationary points of $\mathcal S$ in the
-    complex-$\theta$ plane. In this example,
-    $\Sigma=\{\blacklozenge,\bigstar,\blacktriangle,\blacktriangledown,\bullet,\blacksquare\}$
-    and $\Sigma_0=\{\blacklozenge,\blacktriangledown\}$. Symmetries exist
+    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.
+    \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
+    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
@@ -181,18 +191,35 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of
   } \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
+space $\tilde\Omega$ containing $\Omega$. In many cases this is accomplished by
+simply 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 $\mathbb
+C^N$, 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
+therefore admit a hermitian metric and a symplectic structure. In the extended
+complex phase 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
 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 dz\,e^{-\beta\mathcal S(z)}
+  Z(\beta)=\oint_\Omega ds\,e^{-\beta\mathcal S(s)}
 \end{equation}
-For the moment this translation has only changed some of our symbols from
+For the moment this translation has only changed one of our symbols from
 \eref{eq:partition.function}, but conceptually it is very 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 $\Omega$.
+contour than our initial phase space $\Omega$. This is illustrated in
+Fig.~\ref{fig:contour.deformation}.
 
 \begin{figure}
   \hspace{5pc}
@@ -218,7 +245,7 @@ contour than our initial phase space $\Omega$.
     \textbf{Right:} A funny deformation of the contour in which pieces have been
     pinched off to infinity. So long as no poles have been crossed, even this
     is legal.
-  }
+  } \label{fig:contour.deformation}
 \end{figure}
 
 What contour properties are desirable? Consider the two main motivations cited
@@ -236,16 +263,16 @@ once, courtesy of Morse 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.
 There is one thimble $\mathcal J_\sigma$ associated with each of the stationary
-points $\sigma\in\Sigma$ of the action, and it is defined by all points that
-approach the stationary point $z_\sigma$ under gradient descent on
+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
 $\operatorname{Re}\beta\mathcal S$.
 
 Thimbles guarantee convergent integrals by construction: the value of
 $\operatorname{Re}\beta\mathcal S$ is bounded from below on the thimble $\mathcal J_\sigma$ by its value
-$\operatorname{Re}\beta\mathcal S(z_\sigma)$ at the stationary point,
+$\operatorname{Re}\beta\mathcal S(s_\sigma)$ at the stationary point,
 since all other points on the thimble must descend to reach it. And, as we will
-see in a moment, thimbles guarantee constant phase for the integrand as well, a
-result of the underlying complex geometry of the problem.
+see in the following subsection, thimbles guarantee constant phase for the
+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
@@ -257,9 +284,9 @@ places at infinity (see Fig.~\ref{fig:thimble.homology}). The less simply stated
 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
-is safe for any contour to end up if it wants its integral to converge, since
+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
-integrand vanishes exponentially. The relative homology group
+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
 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
@@ -298,15 +325,40 @@ homology group, and therefore any contour can be represented by some
 composition of thimbles! There is even a systematic way to determine the
 contribution from each thimble: for the stationary point $\sigma\in\Sigma$, let
 $\mathcal K_\sigma$ be its \emph{antithimble}, defined by all points brought to
-$z_\sigma$ by gradient \emph{ascent} (and representing an element of the
+$s_\sigma$ by gradient \emph{ascent} (and representing an element of the
 relative homology group $H_N(\tilde\Omega,\tilde\Omega_{-T})$). Then each
 thimble $\mathcal J_\sigma$ contributes to the contour with a weight given by
 its intersection pairing $n_\sigma=\langle\mathcal C,\mathcal K_\sigma\rangle$.
 
+\begin{figure}
+  \hspace{5pc}
+  \includegraphics{figs/thimble_orientation_1.pdf}\hfill
+  \includegraphics{figs/thimble_orientation_2.pdf}\hfill
+  \includegraphics{figs/thimble_orientation_3.pdf}
+
+  \caption{
+    The behavior of thimble contours near $\arg\beta=0$ for real actions. In all
+    pictures, green arrows depict a canonical orientation of the thimbles
+    relative to the real axis, while purple arrows show the direction of
+    integration implied by the orientation. \textbf{Left:} $\arg\beta=-0.1$. To
+    progress from left to right, one must follow the thimble from the minimum
+    $\mbox{\ding{117}}$ in the direction implied by its orientation, and then
+    follow the thimble from the maximum ${\mbox{\ding{116}}}$ \emph{against} the
+    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
+    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,
+    $\mathcal C=\mathcal J_{\mbox{\ding{117}}}+\mathcal J_{\mbox{\ding{116}}}$.
+  } \label{fig:thimble.orientation}
+\end{figure}
+
 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}dz\,e^{-\beta\mathcal S(z)}.
+  Z(\beta)=\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
@@ -332,42 +384,18 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}.
     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
     darker color, is the same as it was in Fig.~\ref{fig:thimble.homology}.
-    \textbf{Center:} $\arg\beta=1.336$. The thimble $\mathcal J_\blacklozenge$
-    intersects the stationary point $\blacktriangle$ and its thimble, leading
+    \textbf{Center:} $\arg\beta=1.336$. The thimble $\mathcal J_{\mbox{\ding{117}}}$
+    intersects the stationary point ${\mbox{\ding{115}}}$ and its thimble, leading
     to a situation where the contour is not easily defined using thimbles. This
     is a Stokes point. \textbf{Right:} $\arg\beta=1.496$. The Stokes point has passed,
     and the collection of thimbles necessary to produce the path has changed:
-    now $\mathcal J_\blacktriangle$ must be included. Notice that in this
+    now $\mathcal J_{\mbox{\ding{115}}}$ must be included. Notice that in this
     figure, because of the symmetry of the pure models, the thimble $\mathcal
-    J_\blacksquare$ also experiences a Stokes point, but this does not result
+    J_{\mbox{\ding{110}}}$ also experiences a Stokes point, but this does not result
     in a change to the path involving that thimble.
   } \label{fig:1d.stokes}
 \end{figure}
 
-\begin{figure}
-  \hspace{5pc}
-  \includegraphics{figs/thimble_orientation_1.pdf}\hfill
-  \includegraphics{figs/thimble_orientation_2.pdf}\hfill
-  \includegraphics{figs/thimble_orientation_3.pdf}
-
-  \caption{
-    The behavior of thimble contours near $\arg\beta=0$ for real actions. In all
-    pictures, green arrows depict a canonical orientation of the thimbles
-    relative to the real axis, while purple arrows show the direction of
-    integration implied by the orientation. \textbf{Left:} $\arg\beta=-0.1$. To
-    progress from left to right, one must follow the thimble from the minimum
-    $\blacklozenge$ in the direction implied by its orientation, and then
-    follow the thimble from the maximum $\blacktriangledown$ \emph{against} the
-    direction implied by its orientation, from top to bottom. Therefore,
-    $\mathcal C=\mathcal J_\blacklozenge-\mathcal J_\blacktriangledown$.
-    \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
-    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,
-    $\mathcal C=\mathcal J_\blacklozenge+\mathcal J_\blacktriangledown$.
-  } \label{fig:thimble.orientation}
-\end{figure}
 
 The prevalence (or not) of Stokes points in a given continuation, and whether
 those that do appear affect the weights of stationary points of interest, is a
@@ -682,7 +710,7 @@ for those points to have the same imaginary action. However, this is not a
 sufficient condition. This can be seen in Fig.~\ref{fig:4.spin}, which shows
 the thimbles of the circular 6-spin model. The argument of $\beta$ has been
 chosen such that the stationary points marked by $\clubsuit$ and
-$\blacktriangle$ have exactly the same imaginary energy, and yet they do not
+${\mbox{\ding{115}}}$ have exactly the same imaginary energy, and yet they do not
 share a thimble.
 
 \begin{figure}
@@ -692,8 +720,8 @@ share a thimble.
   \caption{
   Some thimbles of the circular 6-spin model, where the argument of $\beta$ has
   been chosen such that the imaginary parts of the action at the stationary
-  points $\clubsuit$ and $\blacktriangle$ are exactly the same (and, as a
-  result of conjugation symmetry, the points $\bigstar$ and $\blacksquare$).
+  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}
 
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