From b48a7192884b2302bf2f2280b22ae991262ba03f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 12 Apr 2022 16:34:29 +0200
Subject: Mork small changes.

---
 stokes.tex | 141 +++++++++++++++++++++++++++++--------------------------------
 1 file changed, 68 insertions(+), 73 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index cc747d9..d9feafe 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -35,12 +35,12 @@
   theory. Topological changes, which occur at so-called Stokes points, proliferate when the
  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
+  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:
   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
+  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
   the topological nature of saddles.
@@ -1127,7 +1127,7 @@ The $p$-spin spherical models are defined by the action
 \begin{equation} \label{eq:p-spin.hamiltonian}
   \mathcal S(x)=\sum_{p=2}^\infty a_p\mathcal S_p(x)
 \end{equation}
-which is a sum of the `pure' actions
+which is a sum of the `pure' $p$-spin actions
 \begin{equation} \label{eq:pure.p-spin.hamiltonian}
   \mathcal S_p(x)=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
 \end{equation}
@@ -1135,7 +1135,7 @@ The variables $x\in\mathbb R^N$ are constrained to lie on the sphere $x^2=N$,
 making the model $D=N-1$ dimensional. The couplings $J$ form totally symmetric
 $p$-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 $a$.
+$a_i=\delta_{ip}$, while the mixed have some more complicated set of coefficients $a$.
 
 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
@@ -1147,7 +1147,7 @@ The projection operator onto the tangent space of this manifold is given by
   P=I-\frac{zz^\dagger}{|z|^2},
 \end{equation}
 where indeed $Pz=z-z|z|^2/|z|^2=0$ and $Pz'=z'$ for any $z'$ orthogonal to $z$.
-When studying stationary points, the constraint can be added to the action in
+When studying stationary points, the constraint can be added to the action
 using a Lagrange multiplier $\mu$ by writing
 \begin{equation}
   \tilde\mathcal S(z)=\mathcal S(z)-\frac\mu2(z^Tz-N)
@@ -1158,8 +1158,8 @@ The gradient of the constraint is simple with $\partial g=z$, and \eqref{eq:mult
   =\frac1Nz^T\partial\mathcal S
   =\sum_{p=2}^\infty a_pp\frac{\mathcal S_p(z)}N
 \end{equation}
-which for the pure $p$-spin in particular implies that $\mu=p\epsilon$ for
-specific energy $\epsilon$.
+For the pure $p$-spin in particular this implies that $\mu=p\epsilon$ for
+specific energy $\epsilon=\mathcal S_p/N$.
 
 \subsection{2-spin}
 \label{subsec:2-spin}
@@ -1172,7 +1172,7 @@ The Hamiltonian of the pure $2$-spin model is defined by
 \begin{equation}
   \mathcal S_2(z)=\frac12z^TJz.
 \end{equation}
-$J$ is generically diagonalizable. In a diagonal basis,
+where the matrix $J$ is generically diagonalizable. In a diagonal basis,
 $J_{ij}=\lambda_i\delta_{ij}$. Then $\partial_i H=\lambda_iz_i$. We will
 henceforth assume to be working in this basis. The constrained action is
 \begin{equation}
@@ -1190,19 +1190,19 @@ zero. In the direction in question,
   \frac1N\frac12\lambda_iz_i^2=\epsilon=\frac12\lambda_i,
 \end{equation}
 whence $z_i=\pm\sqrt{N}$. Thus there are $2N$ stationary points, each
-corresponding to $\pm$ the cardinal directions in the diagonalized basis. The
+corresponding to $\pm$ the cardinal directions on the sphere in the diagonalized basis. The
 energy at each stationary point is real if the couplings are real, and
 therefore there are no complex stationary points in the ordinary 2-spin model.
 
 Imagine for a moment that the coupling are allowed to be complex, giving the
-stationary points of the 2-spin complex energies and therefore potentially
+stationary points of the model complex energies and therefore potentially
 interesting thimble structure. Generically, the eigenvalues of the coupling
 matrix will have distinct imaginary parts, and there will be no Stokes lines.
 Suppose that two stationary points are brought to the same imaginary energy by
 some continuation; without loss of generality, assume these are associated with
 the first and second cardinal directions. Since the gradient is proportional to
 $z$, any components that are zero at some time will be zero at all times. The
-upward flow dynamics for the components of interest assuming all others are
+gradient flow dynamics for the two components of interest assuming all others are
 zero are
 \begin{equation}
   \dot z_1
@@ -1218,7 +1218,7 @@ stationary points are at real $z$, we make this restriction, and find
 \end{equation}
 Therefore $z_2/z_1=e^{\Delta t}$, with $z_1^2+z_2^2=N$ as necessary.  Depending
 on the sign of $\Delta$, $z$ flows from one stationary point to the other over
-infinite time. This is a Stokes line, and establishes that any two stationary
+infinite time. This is a Stokes line, and establishes that any two distinct stationary
 points in the 2-spin model with the same imaginary energy will possess one.
 These trajectories are plotted in Fig.~\ref{fig:two-spin}.
 
@@ -1246,13 +1246,12 @@ in the complex plane, analytic continuation can be made without any fear of
 running into Stokes points. Starting from real, large $\beta$, making an
 infinitesimal phase rotation into the complex plane results in a decomposition
 into thimbles where that of each stationary point is necessary, because all
-stationary points are real. The curvature of the action at the stationary
-points lying at $z_i=\delta_{ik}$ in the $j$th direction is given by
-$\lambda_k-\lambda_j=2(\epsilon_k-\epsilon_k)$. Therefore the generic case of
+stationary points are real and their antithimbles all intersect the real sphere. The curvature of the action at the stationary
+point lying at $z_i=\sqrt N\delta_{ik}$ in the $j$th direction is given by
+$\lambda_k-\lambda_j=2(\epsilon_k-\epsilon_j)$. Therefore the generic case of
 $N$ distinct eigenvalues of the coupling matrix leads to $2N$ stationary points
 with $N$ distinct energies, two at each index from $0$ to $D=N-1$. Starting with
-the expression \eqref{eq:real.thimble.partition.function} valid for the
-partition function contribution from the thimble of a real stationary point, we
+the expression \eqref{eq:real.thimble.partition.function}, we
 have
 \begin{equation}
   \eqalign{
@@ -1269,7 +1268,7 @@ where $\epsilon_k$ is the energy of the twin stationary points of index $k$. In
 \begin{equation} \fl
   \eqalign{
     \overline Z
-    &=2\int d\epsilon\,\rho(\epsilon)\exp\left\{
+    &\simeq2\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\} \\
     &=2\int d\epsilon\,\rho(\epsilon)e^{Nf(\epsilon)}
@@ -1327,7 +1326,7 @@ to zero. This happens for
 or for $|\beta|=1$. Here the sum of contributions from thimbles near the
 maximum again becomes coherent, because the period of oscillations in
 $\epsilon$ diverges at the maximum. These conditions correspond precisely to
-the phase boundaries found for the density of zeros in the 2-spin model found previously using other methods
+the phase boundaries of the density of zeros in the 2-spin model found previously using other methods
 \cite{Obuchi_2012_Partition-function, Takahashi_2013_Zeros}.
 
 We've seen that even in the 2-spin model, which is not complex, making a
@@ -1345,7 +1344,7 @@ continuation.
 \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
+previous work \cite{Kent-Dobias_2021_Complex}. Here, we will review the
 method and elaborate on some of the results relevant to analytic continuation.
 
 The complexity of the real $p$-spin models has been studied extensively, and is even known rigorously \cite{Auffinger_2012_Random}. If $\mathcal N(\epsilon)$ is the number of stationary points with specific energy $\epsilon$, then the complexity is defined by
@@ -1372,7 +1371,7 @@ below which there are exponentially many minima
 compared to saddles, and above which vice versa. This threshold persists in a
 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
+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{subsec:stationary.hessian}. The variance of the $p$-spin hessian without
 shift is
@@ -1440,7 +1439,7 @@ superfield
   \phi(1)=z+\bar\theta(1)\eta+\gamma\theta(1)+\hat z\bar\theta(1)\theta(1)
 \end{equation}
 and its measure $d\phi=dz\,d\hat z\,d\eta\,d\gamma$.
-Then the expression for the number of stationary points can be written in a very compact form, as
+Then the expression for the number of stationary points can be written in a compact form, as
 \begin{equation}
   \mathcal N=\int d\phi^*d\phi\,\exp\left\{
     \int d1\,\operatorname{Re}
@@ -1571,7 +1570,7 @@ opposite sign of the imaginary part of $u$.
 \end{figure}
 
 Contours of this complexity for the pure 3-spin are plotted in
-Fig.~\ref{fig:p-spin.complexity} for pure and imaginary energy. The thick black
+Fig.~\ref{fig:p-spin.complexity} for pure real and imaginary energy. The thick black
 line shows the contour of zero complexity, where stationary points are no
 longer found at large $N$. As the magnitude of the imaginary part of the spin
 taken greater, more stationary points are found, and at a wider array of
@@ -1601,7 +1600,7 @@ have \cite{Bezout_1779_Theorie}.
 Something more interesting is revealed if we zoom in on the complexity around
 the ground state, shown in Fig.~\ref{fig:ground.complexity}. Here, the region
 where most stationary points have a gapped hessian is shaded. The line
-$\epsilon_\mathrm{gap}$ separating gapped from ungapped distribution corresponds
+$\epsilon_\mathrm{gap}$ separating gapped from ungapped spectra corresponds
 to the threshold energy $\epsilon_\mathrm{th}$ in the limit of $Y\to0$.  Above
 the threshold, the limit of the complexity as $Y\to0$ (or
 equivalently $r\to1$) also approaches the real complexity, plotted under the
@@ -1619,7 +1618,7 @@ we shall see more explicitly in the next section. Second, there is only a small
 collection of stationary points that appear with positive complexity and a
 gapped spectrum: the small region in Fig.~\ref{fig:ground.complexity} that is
 both to the right of the thick line and brightly shaded. We suspect that these
-are the only stationary points that are somewhat protected from participation
+are the only stationary points that have any hope of avoiding participation
 in Stokes points.
 
 \subsection{Pure \textit{p}-spin: where are my neighbors?}
@@ -1635,12 +1634,12 @@ In this section, we begin to address the problem heuristically by instead
 asking: if you are at a stationary point, where are your neighbors? The
 stationary points geometrically nearest to a given stationary point should make
 up the bulk of its adjacent points in the sense of being susceptible to Stokes
-points. The distribution of these near neighbors in the complex plane therefore
+points. The distribution of these near neighbors in the complex configuration space therefore
 gives a sense of whether many Stokes lines should be expected, and when.
 
 To determine this, we perform the same Kac--Rice procedure as in the previous
-section, but now with two probe points, or replicas of the system. The simplify
-things somewhat, we will examine only the case where the second probe is
+section, but now with two probe points, or replicas, of the system. The simplify
+things somewhat, we will examine the case where the only second probe is
 complex; the first probe will be on the real sphere. The number of stationary
 points with given energies $\epsilon_1\in\mathbb R$ and $\epsilon_2\in\mathbb C$ are, in the superfield formulation,
 \begin{equation}
@@ -1721,7 +1720,7 @@ They can be simplified somewhat by examination of the real two-replica problem.
 There, all bilinear products involving fermionic fields from different
 replicas, like $\eta_1^T\eta_2$, vanish. This is related to the influence of
 the relative position of the two replicas to their spectra, with the vanishing
-being equivalent to having no influence, e.g., the value of the determinant at
+being equivalent to having no influence, i.e., the value of the determinant at
 each stationary point is exactly what it would be in the one-replica problem
 with the same invariants, e.g., energy and radius. Making this ansatz, the
 equations can be solved for the remaining 5 bilinear products, eliminating all
@@ -1750,8 +1749,8 @@ Define $\theta_{xx}$ as the angle between $x_1$ and $x_2$. Then $x_1^Tx_2=\|x_1\
 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}) \\
-          &=2(1-\sqrt{1-|y_2|^2}\cos\theta_{xx}-i|y_2|\cos\theta_{xy})
+    \gamma\Delta&=2-2x_1^Tx_2-2ix_1^Ty_2=2(1-\|x_2\|\cos\theta_{xx}-i\|y_2\|\cos\theta_{xy}) \\
+          &=2(1-\sqrt{1-\|y_2\|^2}\cos\theta_{xx}-i\|y_2\|\cos\theta_{xy})
   }
 \end{equation}
 where $\theta_{xy}$ is the angle between $x_1$ and $y_2$.
@@ -1785,12 +1784,12 @@ points nearby a given real stationary point look like? We think this is a
 relevant question for the tendency for Stokes lines, for the following reason.
 To determine whether two given stationary points, when tuned to have the same
 imaginary energy, will share a Stokes line, one needs to solve what is known as
-the global connection problem. As we have seen, this as a question of a kind of
-adjacency: two points will \emph{not} share a Stokes line if a third intervenes
-with its thimble between them. We reason that the number of `adjacent'
+the global connection problem. As we have seen in \S\ref{subsec:stokes.conditions}, this as a question of a kind of
+topological adjacency: two points will \emph{not} share a Stokes line if a third intervenes
+with its thimble between them. We reason that the number of adjacent
 stationary points of a given stationary point for a generic function in $D$
-complex dimensions scales linearly with $D$. Therefore, if the collection of
-nearest neighbors has a nonzero complexity, e.g., scales \emph{exponentially}
+complex dimensions scales algebraically with $D$. Therefore, if the collection of
+nearest neighbors has a nonzero complexity, i.e., scales \emph{exponentially}
 with $D$, crowding around the stationary point in question, then these might be
 expected to overwhelm the possible adjacencies, and so doing simplify the
 problem of determining the properties of the true adjacencies. Until the
@@ -1806,7 +1805,7 @@ point sits in the complex configuration space near another,
 $\operatorname{Re}\gamma$ can be related to the angle $\varphi$ made between
 the vector separating these two points and the real configuration space as
 \begin{equation}
-  \varphi=\arctan\sqrt{\frac{1+\operatorname{Re}\gamma}{1-\operatorname{Re}\gamma}}
+  \varphi=\arctan\sqrt{\frac{1-\operatorname{Re}\gamma}{1+\operatorname{Re}\gamma}}
 \end{equation}
 Having concluded that the most populous neighbors are confined to real $\gamma$, we will make use of this angle instead of $\gamma$, which has a more direct geometric interpretation.
 
@@ -1847,7 +1846,7 @@ above and below the threshold, one finds a quickly-converging limit of
 $(\Sigma^{(2)}/\Sigma_{k\geq2}-1)/\Delta$.
 Above the threshold, these curves converge to a function whose peak is always
 precisely at $45^\circ$, while below they converge to a function with a peak
-that grows linearly with $\Delta^{-1}$. At the threshold, the scaling is
+that grows linearly with $\Delta^{-1}$ at $90^\circ$. At the threshold, the scaling is
 different, and the function approaches a flat function extremely rapidly, as
 $\Delta^3$.
 
@@ -1865,9 +1864,7 @@ $\Delta^3$.
     $\epsilon_1=\epsilon_\mathrm{th}+0.001$ \textbf{Center:}
     $\epsilon_1=\epsilon_\mathrm{th}$ \textbf{Right:}
     $\epsilon=\epsilon_\mathrm{th}-0.001$. All lines have been normalized by
-    the complexity $\Sigma$ of the real 3-spin model at the same energy, or
-    (where relevant) by the complexity $\Sigma_2$ of rank-two saddles of the
-    real 3-spin model.
+    the complexity $\Sigma_{k\geq2}$ of index 2 and greater saddles of the real 3-spin model.
   }
 \end{figure}
 
@@ -1947,7 +1944,7 @@ Fig.~\ref{fig:nearest.properties}.
     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. 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.
+    found at drifts. The complexity of nearest neighbors in the shaded region is $\Sigma_{k\geq2}$, while 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}
 
@@ -1972,7 +1969,7 @@ Stokes lines. Defining
   \mathcal L(t)
   = 1-\frac{\operatorname{Re}[\dot z(z(t))^\dagger z'(t)]}{|\dot z(z(t))||z'(t)|}
 \end{equation}
-where $\dot z$ is the flow at $z$ given by \eqref{eq:flow}, this cost is given by
+where $\dot z(z)$ is the flow at $z$ given by \eqref{eq:flow}, this cost is given by
 \begin{equation} \label{eq:cost}
   \mathcal C=\int_0^1 dt\,\mathcal L(t)
 \end{equation}
@@ -1981,7 +1978,7 @@ whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics.
 Therefore, functions that satisfy $\mathcal C=0$ are time-reparameterized Stokes
 lines.
 
-We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to
+We explicitly computed the gradient and hessian of $\mathcal C$ with respect to
 the parameter vectors $g$. Stokes lines are found or not between points by
 using the Levenberg--Marquardt algorithm starting from $g_i=0$ for all $i$,
 and approximating the cost integral by a finite sum. To sample nearby
@@ -1991,13 +1988,13 @@ the \emph{real} configuration space of the $p$-spin model. Then, a
 saddle-finding routine is run on the complex configuration space in the close
 vicinity of the real saddle, using random initial conditions in a slowly
 increasing radius of the real stationary point. When this process finds a new
-distinct stationary point, it is done. This method of sampling pairs heavily
+distinct stationary point, it is finished. This method of sampling pairs heavily
 biases the statistics we report here in favor of seeing Stokes points.
 
 Once a pair of nearby stationary points has been found, one real and one in the
 complex plane, their energies are used to compute the phase $\theta$ necessary
-to give $\beta$ in order to set their imaginary energies to the same value, the
-necessary conditions for a Stokes line. A straight line (ignoring even the
+to give $\beta$ in order to set their imaginary energies to the same value, a
+necessary condition for a Stokes line. A straight line (ignoring even the
 constraint) is thrown between them and then minimized using the cost function
 \eqref{eq:cost} for some initial $m=5$. Once a minimum is found, $m$ is
 iteratively increased several times, each time minimizing the cost in between,
@@ -2005,15 +2002,13 @@ until $m=20$. If at some point in this process the cost blows up, indicating
 that the solution is running away, the pair is thrown out; this happens
 infrequently. At the end, there are several ways to asses whether a given
 minimized line is a Stokes line: the value of the cost, the integrated
-deviation from the constraint, the integrated deviation from the phase of the
-two stationary points. Among minimized lines these values fall into
-doubly-peaked histograms which well-separated prospective Stokes lines into
+deviation from the constraint, and the integrated deviation from constant phase. Among minimized lines these values fall into
+doubly-peaked histograms that well-separate prospective Stokes lines into
 `good' and `bad' values for the given level of approximation $m$.
 
 One cannot explicitly study the effect of crossing various landmark energies on
 the $p$-spin in the system sizes that were accessible to our study, up to
-around $N=64$, as the presence of, e.g., the threshold energy separating
-saddles from minima is not noticeable until much larger size
+around $N=64$, as the presence of, e.g., the threshold energy, is not noticeable until much larger size
 \cite{Folena_2020_Rethinking}. However, we are
 able to examine the effect of its symptoms: namely, the influence of the
 spectrum of the stationary point in question on the likelihood that a randomly
@@ -2068,20 +2063,20 @@ 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
+After all 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
 evidence indicates that the project is hopeless if ungapped stationary points
 take a significant weight in the partition function, since for these Stokes
 lines proliferate at even small continuation and there is no hope of tracking
 them. However, for gapped stationary points we have seen compelling evidence
-that suggests they will not participate in Stokes points, at least until a
+that suggests they will not participate in Stokes points, at least not until a
 large phase rotation of the parameter being continued. This gives some hope for
 continuation of the low-temperature thermodynamic phase of the $p$-spin, where
-weight is concentrated in precisely these points.
+weight is concentrated in precisely gapped minima.
 
-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
+Recalling our expression \eqref{eq:real.thimble.partition.function} for the single-thimble contribution to the partition
+function expanded to lowest order
+in large $|\beta|$, we can write for
 the $p$-spin after an infinitesimal rotation of $\beta$ into the complex plane
 (before any Stokes points have been encountered)
 \begin{equation}
@@ -2093,7 +2088,7 @@ the $p$-spin after an infinitesimal rotation of $\beta$ into the complex plane
       e^{-\beta\mathcal S(s_\sigma)} \\
     &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k)
       \left(\frac{2\pi}\beta\right)^{D/2}i^k
-      |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta N\epsilon}
+      |\det\operatorname{Hess}\mathcal S(\epsilon,k)|^{-\frac12} e^{-\beta N\epsilon}
   }
 \end{equation}
 where $\mathcal N_\mathrm{typ}(\epsilon,k)$ is the typical number of stationary
@@ -2111,16 +2106,16 @@ terms to the typical partition function
 \end{equation}
 where
 \begin{eqnarray} \fl
-  \overline Z_A
+  Z_A
   \simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k)
     \left(\frac{2\pi}\beta\right)^{D/2}i^k
-    |\det\operatorname{Hess}\mathcal S|^{-\frac12} e^{-\beta N\epsilon}
+    |\det\operatorname{Hess}\mathcal S(\epsilon,k)|^{-\frac12} e^{-\beta N\epsilon}
     =\int d\epsilon\,e^{Nf_A(\epsilon)}
     \\ \fl
-  \overline Z_B
+  Z_B
   \simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
     \left(\frac{2\pi}\beta\right)^{D/2}i^k
-    |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} \\
+    |\det\operatorname{Hess}\mathcal S(\epsilon, k)|^{-\frac12}e^{-\beta N\epsilon} \\
     =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)}
 \end{eqnarray}
 for functions $f_A$ and $f_B$ defined by
@@ -2132,22 +2127,22 @@ for functions $f_A$ and $f_B$ defined by
   &=-\beta\epsilon+\frac12\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta
     +i\frac\pi2P(\lambda<0\mid\epsilon)
 \end{eqnarray}
-and $P(\lambda<0\mid\epsilon)$ is the cumulative probability distribution of the eigenvalues of the spectrum given $\epsilon$,
+and where $P(\lambda<0\mid\epsilon)$ is the cumulative probability distribution of the eigenvalues of the spectrum given $\epsilon$,
 \begin{equation}
   P(\lambda<0\mid\epsilon)=\int_{-\infty}^0 d\lambda'\,\rho(\lambda'\mid\epsilon)
 \end{equation}
 and produces the macroscopic index $k/N$.
 Each integral will be dominated by its value near the maximum of the real part of the exponential argument. Assuming that $\epsilon<\epsilon_\mathrm{th}$, this maximum occurs at
 \begin{equation}
-  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_A=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2}
+  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_A\bigg|_{\epsilon=\epsilon_\mathrm{max}}=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon_\mathrm{max}+\frac12\frac p{p-1}\sqrt{\epsilon_\mathrm{max}^2-\epsilon_\mathrm{th}^2}
 \end{equation}
 \begin{equation}
-  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_B=-\operatorname{Re}\beta-\epsilon
+  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_B\bigg|_{\epsilon=\epsilon_\mathrm{max}}=-\operatorname{Re}\beta-\epsilon_\mathrm{max}
 \end{equation}
 As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can
 only be reliably evaluated with a saddle point when either the period of
 oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore
-expect changes in behavior when $\epsilon=\epsilon_{k=0}$, the ground state energy.
+expect changes in behavior when $\epsilon_\mathrm{max}=\epsilon_{k=0}$, the ground state energy.
 The temperature at which this happens is
 \begin{eqnarray}
   \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_{k=0}+\frac12\frac p{p-1}\sqrt{\epsilon_{k=0}^2-\epsilon_\mathrm{th}^2}\\
@@ -2156,7 +2151,7 @@ The temperature at which this happens is
 which for all $p\geq2$ has $\operatorname{Re}\beta_A\geq\operatorname{Re}\beta_B$.
 Therefore, the emergence of zeros in $Z_A$ does not lead to the emergence of
 zeros in the partition function as a whole, because $Z_B$ still produces a
-coherent result (despite the unknown constant factor $\eta(\epsilon_{k=0})$). It is
+coherent result (despite the unknown constant factor $\tilde\eta(\epsilon_{k=0})$). It is
 only at $\operatorname{Re}\beta_B=-\epsilon_{k=0}$ where both terms contributing to
 the partition function at large $N$ involve incoherent integrals near the
 maximum, and only here where the density of zeros is expected to become
@@ -2187,7 +2182,7 @@ Jacobian of the coordinate transformation over the thimble.
 This zeroth-order analysis for the $p$-spin suggests that analytic continuation
 can be sometimes done despite the presence of a great many complex stationary
 points. In particular, when weight is concentrated in certain minima Stokes
-lines do not appear to interrupt the proceedings. How bad the situation in is
+lines do not appear to interrupt the proceedings. How bad the situation is in
 other regimes, like for smaller $|\beta|$, remains to be seen: our analysis
 cannot tell between the effects of Stokes points changing the contour and the
 large-$|\beta|$ saddle-point used to evaluate the thimble integrals. Taking the
@@ -2201,13 +2196,13 @@ We have reviewed the Picard--Lefschetz technique for analytically continuing
 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.
+who seem to avoid Stokes points, and is likely intractable otherwise.
 
 This has implications for the ability to analytically continue other types of
 theories. For instance, \emph{marginal} phases of glasses, spin glasses, and
 other problems are characterized by concentration in pseudogapped minima. Based
 on the considerations of this paper, we suspect that analytic continuation is
-never possible in such a phase, as Stokes points will always proliferate among
+never tractable in such a phase, as Stokes points will always proliferate among
 even the lowest minima.
 
 It is possible that a statistical theory of analytic continuation could be
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