From aafde5c787252a47633ac8bcc4fbf09a28d22ca1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 Feb 2021 15:09:30 +0100
Subject: Added paragraph describing typical norm of critical points.

---
 bezout.tex | 9 +++++++++
 1 file changed, 9 insertions(+)

diff --git a/bezout.tex b/bezout.tex
index d7f4dc7..9222449 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -402,6 +402,15 @@ $\epsilon$ is varied.
   } \label{fig:desert}
 \end{figure}
 
+In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
+critical points are concentrated at infinite $a$, i.e., at complex vectors with
+very large squared norm. For finite $N$ the expectation value $\langle
+a\rangle$ is likewise finite. By differentiating $\overline{\mathcal N}$ with
+respect to $a$ and normalizing, one has an approximation for the distribution
+of critical points as a function of $a$. The expectation value this yields is
+$\langle a\rangle\propto N^{1/2}+O(N^{-1/2})$. One therefore expects typical
+critical points to have a norm that grows modestly with system size.
+
 These qualitative features carry over to nonzero $\epsilon$. In
 Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$
 close to one for which there are no solutions. When $\kappa=1$---the analytic
-- 
cgit v1.2.3-54-g00ecf


From 9471da284a8c6147c608ed1bc23675a2ec71c8ac Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 15:52:39 +0100
Subject: New discussion of the constraint.

---
 bezout.tex | 18 +++++++++++++-----
 1 file changed, 13 insertions(+), 5 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 9222449..1b9f04e 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -98,11 +98,19 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-We choose to constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to
-preserve the analyticity of $H$. The nonholomorphic constraint also has a
-disturbing lack of critical points nearly everywhere: if $H$ were so
-constrained, then $0=\partial^* H=-p\epsilon z$ would only be satisfied for
-$\epsilon=0$.  
+One might balk at taking the constraint as $z^2=N$---which might be more
+appropriately called a hyperbolic constraint---rather than $|z|^2=N$.  The
+reasoning is twofold. First, at every point $z$ the energy
+\eqref{eq:bare.hamiltonian} has a `radial' gradient of magnitude proportional
+to itself, as $z\cdot\partial H_0=pH_0$. This trivial direction must be removed
+if critical points are to exist a any nonzero energy, and the constraint
+surface $z^2=N$ is the unique surface whose normal is parallel to $z$ and which
+contains the real configuration space as a subspace. Second, taking the
+constraint to be the level set of a holomorphic function means the resulting
+configuration space is a \emph{bone fide} complex manifold, and therefore
+admits easy generalization of the integration techniques referenced above. The
+same cannot be said for the space defined by $|z|^2=N$, which is topologically
+the $(2N-1)$-sphere and cannot admit a complex structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
cgit v1.2.3-54-g00ecf


From 145d4716478344e6ec7748c5df814e85dee56a43 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 15:58:49 +0100
Subject: Working tweak.s

---
 bezout.tex | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 1b9f04e..ff5a840 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -102,15 +102,16 @@ One might balk at taking the constraint as $z^2=N$---which might be more
 appropriately called a hyperbolic constraint---rather than $|z|^2=N$.  The
 reasoning is twofold. First, at every point $z$ the energy
 \eqref{eq:bare.hamiltonian} has a `radial' gradient of magnitude proportional
-to itself, as $z\cdot\partial H_0=pH_0$. This trivial direction must be removed
-if critical points are to exist a any nonzero energy, and the constraint
-surface $z^2=N$ is the unique surface whose normal is parallel to $z$ and which
-contains the real configuration space as a subspace. Second, taking the
-constraint to be the level set of a holomorphic function means the resulting
-configuration space is a \emph{bone fide} complex manifold, and therefore
-admits easy generalization of the integration techniques referenced above. The
-same cannot be said for the space defined by $|z|^2=N$, which is topologically
-the $(2N-1)$-sphere and cannot admit a complex structure.
+to the energy, as $z\cdot\partial H_0=pH_0$. This trivial direction must be
+removed if critical points are to exist a any nonzero energy, and the
+constraint surface $z^2=N$ is the unique surface whose normal is parallel to
+$z$ and which contains the configuration space of the real $p$-spin model as a
+subspace. Second, taking the constraint to be the level set of a holomorphic
+function means the resulting configuration space is a \emph{bone fide} complex
+manifold, and therefore permits easy generalization of the integration
+techniques referenced above. The same cannot be said for the space defined by
+$|z|^2=N$, which is topologically the $(2N-1)$-sphere and cannot admit a
+complex structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
cgit v1.2.3-54-g00ecf


From 55409dc28d44271a915e9197a423531a06a17d4a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 16:09:33 +0100
Subject: More working changes to constraint discussion.

---
 bezout.tex | 27 +++++++++++++--------------
 1 file changed, 13 insertions(+), 14 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index ff5a840..3a02f51 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -98,20 +98,19 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-One might balk at taking the constraint as $z^2=N$---which might be more
-appropriately called a hyperbolic constraint---rather than $|z|^2=N$.  The
-reasoning is twofold. First, at every point $z$ the energy
-\eqref{eq:bare.hamiltonian} has a `radial' gradient of magnitude proportional
-to the energy, as $z\cdot\partial H_0=pH_0$. This trivial direction must be
-removed if critical points are to exist a any nonzero energy, and the
-constraint surface $z^2=N$ is the unique surface whose normal is parallel to
-$z$ and which contains the configuration space of the real $p$-spin model as a
-subspace. Second, taking the constraint to be the level set of a holomorphic
-function means the resulting configuration space is a \emph{bone fide} complex
-manifold, and therefore permits easy generalization of the integration
-techniques referenced above. The same cannot be said for the space defined by
-$|z|^2=N$, which is topologically the $(2N-1)$-sphere and cannot admit a
-complex structure.
+One might balk at the constraint $z^2=N$---which could appropriately be called
+a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$.  The reasoning
+is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian}
+has a `radial' gradient of magnitude proportional to the energy, since
+$z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical
+points are to exist a any nonzero energy, and the constraint surface $z^2=N$ is
+the unique surface whose normal is parallel to $z$ and which contains the
+configuration space of the real $p$-spin model as a subspace. Second, taking
+the constraint to be the level set of a holomorphic function means the
+resulting configuration space is a \emph{bone fide} complex manifold, and
+therefore permits easy generalization of the integration techniques referenced
+above. The same cannot be said for the space defined by $|z|^2=N$, which is
+topologically the $(2N-1)$-sphere and cannot admit a complex structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
cgit v1.2.3-54-g00ecf


From 823e1cb5e7ab98f7ebda8b8517e427241ec427ce Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 Mar 2021 16:09:55 +0100
Subject: Spelling mistake.

---
 bezout.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/bezout.tex b/bezout.tex
index 3a02f51..7cff5f1 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -103,7 +103,7 @@ a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$.  The reasoning
 is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian}
 has a `radial' gradient of magnitude proportional to the energy, since
 $z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical
-points are to exist a any nonzero energy, and the constraint surface $z^2=N$ is
+points are to exist at any nonzero energy, and the constraint surface $z^2=N$ is
 the unique surface whose normal is parallel to $z$ and which contains the
 configuration space of the real $p$-spin model as a subspace. Second, taking
 the constraint to be the level set of a holomorphic function means the
-- 
cgit v1.2.3-54-g00ecf


From 57d6f4d6fd0a7fdb738b91354de063f6a74847d1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 12:30:49 +0100
Subject: More rewriting of constraint reasoning.

---
 bezout.tex | 34 ++++++++++++++++++++++------------
 1 file changed, 22 insertions(+), 12 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 7cff5f1..efe9b62 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -99,18 +99,28 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
 One might balk at the constraint $z^2=N$---which could appropriately be called
-a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$.  The reasoning
-is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian}
-has a `radial' gradient of magnitude proportional to the energy, since
-$z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical
-points are to exist at any nonzero energy, and the constraint surface $z^2=N$ is
-the unique surface whose normal is parallel to $z$ and which contains the
-configuration space of the real $p$-spin model as a subspace. Second, taking
-the constraint to be the level set of a holomorphic function means the
-resulting configuration space is a \emph{bone fide} complex manifold, and
-therefore permits easy generalization of the integration techniques referenced
-above. The same cannot be said for the space defined by $|z|^2=N$, which is
-topologically the $(2N-1)$-sphere and cannot admit a complex structure.
+a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning
+behind the choice is twofold.
+
+First, we seek draw conclusions from our model that would be applicable to
+generic holomorphic functions without any symmetry.  Samples of $H_0$ nearly
+provide this, save for a single anomaly: the value of the energy and its
+gradient at any point $z$ correlate along the $z$ direction, with
+$\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and
+$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$.  Besides being a
+spurious correlation, in each sample there is also a `radial' gradient of
+magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
+anomalous direction must be neglected if we are to draw conclusions about
+generic functions, and the constraint surface $z^2=N$ is the unique surface
+whose normal is parallel to $z$ and which contains the configuration space of
+the real $p$-spin model as a subspace.
+
+Second, taking the constraint to be the level set of a holomorphic function
+means the resulting configuration space is a \emph{bone fide} complex manifold,
+and therefore permits easy generalization of the integration techniques
+referenced above. The same cannot be said for the space defined by $|z|^2=N$,
+which is topologically the $(2N-1)$-sphere and cannot admit a complex
+structure.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
-- 
cgit v1.2.3-54-g00ecf


From f7df7235a9b8d44ab47a61aec32b215c88549fa1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 12:40:58 +0100
Subject: Spot changes.

---
 bezout.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index efe9b62..c0ecff7 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -103,11 +103,11 @@ a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning
 behind the choice is twofold.
 
 First, we seek draw conclusions from our model that would be applicable to
-generic holomorphic functions without any symmetry.  Samples of $H_0$ nearly
+generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
 provide this, save for a single anomaly: the value of the energy and its
 gradient at any point $z$ correlate along the $z$ direction, with
 $\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and
-$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$.  Besides being a
+$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a
 spurious correlation, in each sample there is also a `radial' gradient of
 magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
 anomalous direction must be neglected if we are to draw conclusions about
@@ -132,7 +132,7 @@ equations of degree $p-1$, to which one must add the constraint.  In this sense
 this study also provides a complement to the work on the distribution of zeroes
 of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
 and $p\to\infty$.  We see from \eqref{eq:polynomial} that at any critical
-point, $\epsilon=H/N$, the average energy.
+point $\epsilon=H/N$, the average energy.
 
 Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a
 critical point of $\operatorname{Im}H$. The number of critical points of $H$ is
@@ -187,7 +187,7 @@ $(\partial\partial H)^\dagger\partial\partial H$.
 
 The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give
 the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \,
-\log \mathcal N_J$, a calculation that involves the replica trick. In most the
+\log \mathcal N_J$, a calculation that involves the replica trick. In most of the
 parameter-space that we shall study here, the \emph{annealed approximation} $N
 \Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is
 exact. 
-- 
cgit v1.2.3-54-g00ecf


From 9aa887b6e9cf240de5efbd26411d6a2aad834366 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 15:18:53 +0100
Subject: Started converting a to r where relevant.

---
 bezout.tex         |  83 +++++++++++++++++++++++++++++------------------------
 fig/complexity.pdf | Bin 12732 -> 12775 bytes
 fig/desert.pdf     | Bin 14593 -> 14218 bytes
 3 files changed, 46 insertions(+), 37 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index c0ecff7..eaa98ce 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -122,6 +122,18 @@ referenced above. The same cannot be said for the space defined by $|z|^2=N$,
 which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
+A consequence of the constraint is that the model's configuration space is not
+compact, nor is its energy bounded. This is not necessarily a problem, as many
+related problems have similar properties but are concerned with subspaces on
+which the energy is bounded. (In fact, identifying the appropriate subspace on
+which to define one's model often requires the study of critical points in the
+whole space.) Where it might be a problem, we introduce the additional
+constraint $|z|^2\leq Nr^2$. The resulting configuration space is a complex
+manifold with boundary. We shall see that the `radius' $r$ proves an insightful
+knob in our present problem, revealing structure as it is varied. Note
+that---combined with the constraint $z^2=N$---taking $r=1$ reduces the problem
+to that of the ordinary $p$-spin.
+
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
   \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}}
@@ -145,7 +157,7 @@ of $2N$ real variables. Its number of saddle-points is given by the usual
 Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon)
+    \mathcal N_J&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\hspace{6pc}\times\left|\det\begin{bmatrix}
           \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
@@ -160,7 +172,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
 transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon)
+    \mathcal N_J&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
                 &\hspace{6pc}\times\left|\det\begin{bmatrix}
             \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -203,7 +215,7 @@ and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat
 z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their
 conjugates). The result, to leading order in $N$, is
 \begin{equation} \label{eq:saddle}
-    \overline{\mathcal N}(\kappa,\epsilon)
+    \overline{\mathcal N}(\kappa,\epsilon,r)
         = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)},
 \end{equation}
 where the argument of the exponential is
@@ -352,10 +364,11 @@ geometry problem, and yields
 for $\delta=\kappa a^{-(p-2)}$.
 
 Given $\rho$, the integral in \eqref{eq:free.energy.a} may be preformed for
-arbitrary $a$. The resulting expression is maximized for $a=\infty$ for all
-values of $\kappa$ and $\epsilon$. Taking this saddle gives
+arbitrary $a$. The resulting expression is maximized for $a=r^2$ for all
+values of $\kappa$ and $\epsilon$. Evaluating the complexity at this saddle, in
+the limit of unbounded spins, gives
 \begin{equation} \label{eq:bezout}
-  \log\overline{\mathcal N}(\kappa,\epsilon)
+  \lim_{r\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,r)
   =N\log(p-1).
 \end{equation}
 This is, to this order,  precisely the Bézout bound, the maximum number of
@@ -367,41 +380,34 @@ surprising, since the coefficients of our polynomial equations
 Analogous asymptotic scaling has been found for the number of pure Higgs states
 in supersymmetric quiver theories \cite{Manschot_2012_From}.
 
-More insight is gained by looking at the count as a function of $a$, defined by
-$\overline{\mathcal N}(\kappa,\epsilon,a)=e^{Nf(a)}$.  In the large-$N$ limit,
-this is the cumulative number of critical points, or the number of critical
-points $z$ for which $|z|^2\leq a$. We likewise define the $a$-dependant
-complexity $\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal
-N}(\kappa,\epsilon,a)$
-
 \begin{figure}[htpb]
   \centering
   \includegraphics{fig/complexity.pdf}
   \caption{
     The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    $a=|z|^2=1+y^2$ at several values of $\kappa$. The dashed line shows
-    $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
+    the maximum `radius' $r=|z_{\mathrm{max}}|/\sqrt N$ at several values of
+    $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows
+    $\log(p-1)$.
   } \label{fig:complexity}
 \end{figure}
 
-Everything is analytically tractable for $\epsilon=0$, giving
+For finite $r$, everything is analytically tractable at $\epsilon=0$, giving
 \begin{equation} \label{eq:complexity.zero.energy}
-  \Sigma(\kappa,0,a)
-  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2a^{-2(p-1)}}{1-a^{-2}}\right).
+  \Sigma(\kappa,0,r)
+  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2r^{-4(p-1)}}{1-r^{-4}}\right).
 \end{equation}
-Notice that the limit of this expression as $a\to\infty$ corresponds with
-\eqref{eq:bezout}, as expected. This is plotted as a function of $a$ for
+This is plotted as a function of $r$ for
 several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$,
-the complexity goes to negative infinity as $a\to1$, i.e., as the spins are
+the complexity goes to negative infinity as $r\to1$, i.e., as the spins are
 restricted to the reals.  This is natural, given that the $y$ contribution to
-the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(a-1)$ with volume
-$\sim(a-1)^N$.  However, when the result is analytically continued to
+the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(r^2-1)$ with volume
+$\sim(r^2-1)^N$.  However, when the result is analytically continued to
 $\kappa=1$ (which corresponds to real $J$) something novel occurs: the
-complexity has a finite value at $a=1$. Since the $a$-dependence gives a
+complexity has a finite value at $r=1$. Since the $r$-dependence gives a
 cumulative count, this implies a $\delta$-function density of critical points
 along the line $y=0$.  The number of critical points contained within is
 \begin{equation}
-  \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a)
+  \lim_{r\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,r)
   = \frac12N\log(p-1),
 \end{equation}
 half of \eqref{eq:bezout} and corresponding precisely to the number of critical
@@ -414,27 +420,30 @@ $\epsilon$ is varied.
   \centering
   \includegraphics{fig/desert.pdf}
   \caption{
-    The minimum value of $a$ for which the complexity is positive as a function
-    of (real) energy $\epsilon$  for the 3-spin model at several values of
-    $\kappa$.
+    The value of `radius' $r$ for which $\Sigma(\kappa,\epsilon,r)=0$ as a
+    function of (real) energy per spin $\epsilon$  for the 3-spin model at
+    several values of $\kappa$. Above each line the complexity is positive and
+    critical points proliferate, while below it the complexity is negative and
+    critical points are exponentially suppressed. The dotted black lines show
+    the location of the ground and highest exited state energies for the real
+    3-spin model.
   } \label{fig:desert}
 \end{figure}
 
 In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
-critical points are concentrated at infinite $a$, i.e., at complex vectors with
-very large squared norm. For finite $N$ the expectation value $\langle
-a\rangle$ is likewise finite. By differentiating $\overline{\mathcal N}$ with
-respect to $a$ and normalizing, one has an approximation for the distribution
-of critical points as a function of $a$. The expectation value this yields is
-$\langle a\rangle\propto N^{1/2}+O(N^{-1/2})$. One therefore expects typical
+critical points are concentrated at infinite radius, i.e., at complex vectors with
+very large squared norm. For finite $N$ the average radius of critical points is likewise finite. By differentiating $\overline{\mathcal N}$ with
+respect to $r$ and normalizing, one has the distribution
+of critical points as a function of $r$. The average radius this yields is
+$\propto N^{1/4}+O(N^{-3/4})$. One therefore expects typical
 critical points to have a norm that grows modestly with system size.
 
 These qualitative features carry over to nonzero $\epsilon$. In
-Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$
+Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $r$
 close to one for which there are no solutions. When $\kappa=1$---the analytic
 continuation to the real computation---the situation is more interesting. In
 the range of energies where there are real solutions this gap closes, which is
-only possible if the density of solutions diverges at $a=1$.  Another
+only possible if the density of solutions diverges at $r=1$.  Another
 remarkable feature of this limit is that there is still a gap without solutions
 around `deep' real energies where there is no real solution. A moment's thought
 tells us that this is a necessity: otherwise a small perturbation of the $J$s
@@ -463,7 +472,7 @@ is richer than in the real case. In Fig.~\ref{fig:eggs} these are shown in the
 complex-$\epsilon$ plane for several examples. Depending on the parameters, the
 threshold might always come at smaller magnitude than the extremal state, or
 always come at larger magnitude, or cross as a function of complex argument.
-For sufficiently large $a$ the threshold always comes at larger magnitude than
+For sufficiently large $r$ the threshold always comes at larger magnitude than
 the extremal state. If this were to happen in the real case, it would likely
 imply our replica symmetric computation is unstable, since having a ground
 state above the threshold implies a ground state Hessian with many negative
diff --git a/fig/complexity.pdf b/fig/complexity.pdf
index b68f2cf..f9336bb 100644
Binary files a/fig/complexity.pdf and b/fig/complexity.pdf differ
diff --git a/fig/desert.pdf b/fig/desert.pdf
index e19484a..08d8f41 100644
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-- 
cgit v1.2.3-54-g00ecf


From 43ed1805b695086eb1eb7218fc483557a6df82be Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 16:38:17 +0100
Subject: Changed some notation to be more clear.

---
 bezout.tex | 32 +++++++++++++-------------------
 1 file changed, 13 insertions(+), 19 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index eaa98ce..458b448 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -51,7 +51,7 @@ review see \cite{Castellani_2005_Spin-glass}) defined by the energy
   H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
 where $J$ is a symmetric tensor whose elements are real Gaussian variables and
-$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been
+$z\in\mathbb R^N$ is constrained to the sphere $z^Tz=N$. This problem has been
 studied in the algebra \cite{Cartwright_2013_The} and probability literature
 \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.  It has been attacked
 from several angles: the replica trick to compute the Boltzmann--Gibbs
@@ -68,7 +68,7 @@ In this paper we extend the study to complex variables: we shall take
 $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are
 \emph{complex} normal, with $\overline{|J|^2}=p!/2N^{p-1}$ and
 $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The
-constraint remains $z^2=N$.
+constraint remains $z^Tz=N$.
 
 The motivations for this paper are of two types. On the practical side, there
 are indeed situations in which complex variables appear naturally in disordered
@@ -98,27 +98,26 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-One might balk at the constraint $z^2=N$---which could appropriately be called
-a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning
+One might balk at the constraint $z^Tz=N$---which could appropriately be called
+a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$. The reasoning
 behind the choice is twofold.
 
 First, we seek draw conclusions from our model that would be applicable to
 generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
 provide this, save for a single anomaly: the value of the energy and its
 gradient at any point $z$ correlate along the $z$ direction, with
-$\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and
-$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a
+$\overline{H_0\partial_iH_0}\propto \overline{H_0(\partial_iH_0)^*}\propto z_i$. Besides being a
 spurious correlation, in each sample there is also a `radial' gradient of
 magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
 anomalous direction must be neglected if we are to draw conclusions about
-generic functions, and the constraint surface $z^2=N$ is the unique surface
+generic functions, and the constraint surface $z^Tz=N$ is the unique surface
 whose normal is parallel to $z$ and which contains the configuration space of
 the real $p$-spin model as a subspace.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
 and therefore permits easy generalization of the integration techniques
-referenced above. The same cannot be said for the space defined by $|z|^2=N$,
+referenced above. The same cannot be said for the space defined by $z^\dagger z=N$,
 which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
@@ -128,10 +127,10 @@ related problems have similar properties but are concerned with subspaces on
 which the energy is bounded. (In fact, identifying the appropriate subspace on
 which to define one's model often requires the study of critical points in the
 whole space.) Where it might be a problem, we introduce the additional
-constraint $|z|^2\leq Nr^2$. The resulting configuration space is a complex
+constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex
 manifold with boundary. We shall see that the `radius' $r$ proves an insightful
 knob in our present problem, revealing structure as it is varied. Note
-that---combined with the constraint $z^2=N$---taking $r=1$ reduces the problem
+that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem
 to that of the ordinary $p$-spin.
 
 The critical points are of $H$ given by the solutions to the set of equations
@@ -211,8 +210,7 @@ be averaged independently. The $\delta$-functions are converted to exponentials
 by the introduction of auxiliary fields $\hat z=\hat x+i\hat y$.  The average
 of those factors over $J$ can then be performed. A generalized
 Hubbard--Stratonovich allows a change of variables from the $4N$ original
-and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat
-z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their
+and auxiliary fields to eight bilinears defined by $Na=z^\dagger z$, $N\hat a=\hat z^\dagger\hat z$, $N\hat c=\hat z^T\hat z$, $Nb=\hat z^\dagger z$, and $Nd=\hat z^Tz$ (and their
 conjugates). The result, to leading order in $N$, is
 \begin{equation} \label{eq:saddle}
     \overline{\mathcal N}(\kappa,\epsilon,r)
@@ -244,10 +242,7 @@ where $\theta=\frac12\arg\kappa$ and
 \begin{equation}
   C_{\pm}=\frac{a^p(1+p(a^2-1))\mp a^2|\kappa|}{a^{2p}\pm(p-1)a^p(a^2-1)|\kappa|-a^2|\kappa|^2}.
 \end{equation}
-This leaves a single parameter, $a$, which dictates the magnitude of $|z|^2$,
-or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes
-as $a\to1$, where (as we shall see) one recovers known results for the real
-$p$-spin.
+This leaves a single parameter, $a$, which dictates the norm of $z$. 
 
 The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
 the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
@@ -385,9 +380,8 @@ in supersymmetric quiver theories \cite{Manschot_2012_From}.
   \includegraphics{fig/complexity.pdf}
   \caption{
     The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    the maximum `radius' $r=|z_{\mathrm{max}}|/\sqrt N$ at several values of
-    $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows
-    $\log(p-1)$.
+    the maximum `radius' $r$ at several values of $\kappa$. The dashed line
+    shows $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
   } \label{fig:complexity}
 \end{figure}
 
-- 
cgit v1.2.3-54-g00ecf


From 49201233c5d8164f5d1633d13a3442f231e76d3d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 16:38:34 +0100
Subject: New figure that may replace egg plots.

---
 fig/threshold.pdf | Bin 0 -> 10063 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 fig/threshold.pdf

diff --git a/fig/threshold.pdf b/fig/threshold.pdf
new file mode 100644
index 0000000..538d3d8
Binary files /dev/null and b/fig/threshold.pdf differ
-- 
cgit v1.2.3-54-g00ecf


From 875f996cd76d1c534beca1beb2e0e821e3ea84c6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 16:58:08 +0100
Subject: Added draft of appeal letter.

---
 appeal.tex | 74 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 74 insertions(+)
 create mode 100644 appeal.tex

diff --git a/appeal.tex b/appeal.tex
new file mode 100644
index 0000000..d1df16c
--- /dev/null
+++ b/appeal.tex
@@ -0,0 +1,74 @@
+\documentclass[a4paper]{letter}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[
+  colorlinks=true,
+  urlcolor=purple,
+  linkcolor=black,
+  citecolor=black,
+  filecolor=black
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{xcolor}
+\usepackage[style=phys]{biblatex}
+
+\addbibresource{bezout.bib}
+
+\signature{
+  \vspace{-6\medskipamount}
+  \smallskip
+  Jaron Kent-Dobias \& Jorge Kurchan
+}
+
+\address{
+  Laboratoire de Physique\\
+  Ecole Normale Sup\'erieure\\
+  24 rue Lhomond\\ 
+  75005 Paris
+}
+
+\begin{document}
+\begin{letter}{
+  Editorial Office\\
+  Physical Review Letters\\
+  1 Research Road\\
+  Ridge, NY 11961
+}
+
+\opening{To the editors of Physical Review,}
+
+We wish to appeal your decision on our manuscript LZ16835, \emph{Complex
+complex landscapes}, which received a single referee report.
+
+
+We believe that the criticisms that the referee addresses to our paper are
+not entirely justified (and above all, difficult to answer).  We have, however,
+clarified as much as possible the parts that the referee found trying.
+
+The referee is particularly worried that we have cited articles that are not
+themselves sufficiently cited, so we thought that it may be useful at this
+point to propose a set of referees that are beyond suspicion of incompetence or
+uncitedness:
+
+\begin{tabular}{ll}
+  G Ben Arous  & Courant \\ 
+  M Berry      & Bristol\\
+  Daniel Fisher& Stanford\\
+  T Lubensky   & Penn\\
+  M Moore      & Manchester \\
+  E Witten     & IAS Princeton
+\end{tabular}
+
+We have also added  some very first results of the geometric implications for a
+random complex landscape of our calculation, something we plan to expend on in
+a future full article.
+
+
+\closing{Sincerely,}
+
+\vspace{1em}
+
+\end{letter}
+
+\end{document}
-- 
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From 85002f83cae33123e568413f6c5b811d429431f2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 16:58:40 +0100
Subject: Simplified constraint arguement somewhat.

---
 bezout.tex | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 458b448..db14a52 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -106,9 +106,7 @@ First, we seek draw conclusions from our model that would be applicable to
 generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
 provide this, save for a single anomaly: the value of the energy and its
 gradient at any point $z$ correlate along the $z$ direction, with
-$\overline{H_0\partial_iH_0}\propto \overline{H_0(\partial_iH_0)^*}\propto z_i$. Besides being a
-spurious correlation, in each sample there is also a `radial' gradient of
-magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
+$\overline{H_0\partial H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This
 anomalous direction must be neglected if we are to draw conclusions about
 generic functions, and the constraint surface $z^Tz=N$ is the unique surface
 whose normal is parallel to $z$ and which contains the configuration space of
-- 
cgit v1.2.3-54-g00ecf


From 720776cd65074b3eb5edb6a16e672022523bb81c Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 17:07:11 +0100
Subject: Appeal wording.

---
 appeal.tex | 18 ++++++++----------
 1 file changed, 8 insertions(+), 10 deletions(-)

diff --git a/appeal.tex b/appeal.tex
index d1df16c..2cf4757 100644
--- a/appeal.tex
+++ b/appeal.tex
@@ -38,18 +38,16 @@
 
 \opening{To the editors of Physical Review,}
 
-We wish to appeal your decision on our manuscript LZ16835, \emph{Complex
-complex landscapes}, which received a single referee report.
+We wish to appeal your decision on our manuscript \emph{Complex complex
+landscapes}, which received a single referee report.
 
+We believe that the referee's overall criticisms of our paper are not entirely
+justified (and above all, difficult to answer).  We have, however, submitted a
+revised manuscript clarifying the specific aspects that the referee found trying.
 
-We believe that the criticisms that the referee addresses to our paper are
-not entirely justified (and above all, difficult to answer).  We have, however,
-clarified as much as possible the parts that the referee found trying.
-
-The referee is particularly worried that we have cited articles that are not
-themselves sufficiently cited, so we thought that it may be useful at this
-point to propose a set of referees that are beyond suspicion of incompetence or
-uncitedness:
+The referee seems particularly worried that we have cited articles that are not
+themselves sufficiently cited, so we thought it may be useful propose a set of
+referees that are beyond suspicion of incompetence or uncitedness:
 
 \begin{tabular}{ll}
   G Ben Arous  & Courant \\ 
-- 
cgit v1.2.3-54-g00ecf


From ca2c30ef85e8e9e9e42a03127cb23ef8f1b6dfbe Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 17:29:02 +0100
Subject: Wording changes.

---
 bezout.tex | 32 +++++++++++++-------------------
 1 file changed, 13 insertions(+), 19 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index db14a52..f950b8c 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -128,8 +128,7 @@ whole space.) Where it might be a problem, we introduce the additional
 constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex
 manifold with boundary. We shall see that the `radius' $r$ proves an insightful
 knob in our present problem, revealing structure as it is varied. Note
-that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem
-to that of the ordinary $p$-spin.
+that taking $r=1$ reduces the problem to that of the ordinary $p$-spin.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
@@ -143,18 +142,14 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
 and $p\to\infty$.  We see from \eqref{eq:polynomial} that at any critical
 point $\epsilon=H/N$, the average energy.
 
-Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a
-critical point of $\operatorname{Im}H$. The number of critical points of $H$ is
-therefore the same as that of $\operatorname{Re}H$. From each saddle 
-emerge gradient lines of $\operatorname{Re}H$, which are also ones of constant
-$\operatorname{Im}H$ and therefore constant phase.
-
-Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function
-of $2N$ real variables. Its number of saddle-points is given by the usual
-Kac--Rice formula:
+Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also
+one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for
+$x,y\in\mathbb R^N$, $\operatorname{Re}H$ can be considered a real-valued
+function of $2N$ real variables. The number of critical points of $H$ is thus given by the
+usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon,r)
+    \mathcal N&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\hspace{6pc}\times\left|\det\begin{bmatrix}
           \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
@@ -169,7 +164,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
 transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon,r)
+    \mathcal N&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
                 &\hspace{6pc}\times\left|\det\begin{bmatrix}
             \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -183,7 +178,7 @@ transformations through, we have
 \end{equation}
 This gives three equivalent expressions for the determinant of the Hessian: as
 that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix,
-i.e. the norm squared of that of an $N\times N$ complex symmetric matrix.
+or the norm squared of that of an $N\times N$ complex symmetric matrix.
 
 These equivalences belie a deeper connection between the spectra of the
 corresponding matrices. Each positive eigenvalue of the real matrix has a
@@ -195,11 +190,10 @@ $\partial\partial H$, or the distribution of square-rooted eigenvalues of
 $(\partial\partial H)^\dagger\partial\partial H$.
 
 The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give
-the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \,
-\log \mathcal N_J$, a calculation that involves the replica trick. In most of the
-parameter-space that we shall study here, the \emph{annealed approximation} $N
-\Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is
-exact. 
+the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N}$, a calculation
+that involves the replica trick. In most of the parameter-space that we shall
+study here, the \emph{annealed approximation} $N \Sigma \sim \log \overline{
+\mathcal N}$ is exact. 
 
 A useful property of the Gaussian $J$ is that gradient and Hessian at fixed
 $\epsilon$ are statistically independent \cite{Bray_2007_Statistics,
-- 
cgit v1.2.3-54-g00ecf


From 29cae4315ff61a3124e77ff91fe401874e120612 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 15 Mar 2021 15:51:58 +0100
Subject: Big rewrite for clarity.

---
 bezout.tex         | 394 +++++++++++++++++++++++++++--------------------------
 fig/complexity.pdf | Bin 12775 -> 12765 bytes
 fig/desert.pdf     | Bin 14218 -> 14242 bytes
 3 files changed, 202 insertions(+), 192 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index f950b8c..f050707 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -42,11 +42,11 @@
 
 \maketitle
 
-Spin-glasses have long been considered the paradigm of many variable `complex
-landscapes,' a subject that includes neural networks and optimization problems,
-most notably constraint satisfaction \cite{Mezard_2009_Information}.  The most tractable family of these
-are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The} (for a
-review see \cite{Castellani_2005_Spin-glass}) defined by the energy
+Spin-glasses are the paradigm of many-variable `complex landscapes,' a category
+that also includes neural networks and optimization problems like constraint
+satisfaction \cite{Mezard_2009_Information}.  The most tractable family of
+these are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The}
+(for a review see \cite{Castellani_2005_Spin-glass}) defined by the energy
 \begin{equation} \label{eq:bare.hamiltonian}
   H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
@@ -58,11 +58,10 @@ from several angles: the replica trick to compute the Boltzmann--Gibbs
 distribution \cite{Crisanti_1992_The}, a Kac--Rice \cite{Kac_1943_On,
 Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the
 Fadeev--Popov integral) to compute the number of saddle-points of the energy
-function \cite{Crisanti_1995_Thouless-Anderson-Palmer}, and the
-gradient-descent---or more generally Langevin---dynamics staring from a
-high-energy configuration \cite{Cugliandolo_1993_Analytical}. Thanks to the
-simplicity of the energy, all these approaches yield analytic results in the
-large-$N$ limit.
+function \cite{Crisanti_1995_Thouless-Anderson-Palmer}, and gradient-descent
+(or more generally Langevin) dynamics starting from a high-energy configuration
+\cite{Cugliandolo_1993_Analytical}. Thanks to the simplicity of the energy, all
+these approaches yield analytic results in the large-$N$ limit.
 
 In this paper we extend the study to complex variables: we shall take
 $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are
@@ -77,7 +76,7 @@ random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used
 to model black hole horizons in the zero-temperature limit---also have a
 Hamiltonian very close to ours \cite{Anninos_2016_Disordered}.  A second reason
 is that, as we know from experience, extending a real problem to the complex
-plane often uncovers underlying simplicity that is otherwise hidden, sheding
+plane often uncovers underlying simplicity that is otherwise hidden, shedding
 light on the original real problem, e.g., as in the radius of convergence of a
 series.
 
@@ -86,15 +85,16 @@ $2N$-dimensional complex space has turned out to be necessary for correctly
 defining and analyzing path integrals with complex action (see
 \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the
 sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient,
-Scorzato_2016_The}.  In order to do this correctly, the features of landscape
+Scorzato_2016_The}.  In order to do this correctly, features of landscape
 of the action in complex space---like the relative position of its
 saddles---must be understood. Such landscapes are in general not random: here
 we propose to follow the strategy of computer science of understanding the
-generic features of random instances, expecting that this sheds light on the
+generic features of random instances, expecting that this sheds light on
 practical, nonrandom problems.
 
 Returning to our problem, the spherical constraint is enforced using the method
-of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
+of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our constrained
+energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
@@ -102,35 +102,35 @@ One might balk at the constraint $z^Tz=N$---which could appropriately be called
 a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$. The reasoning
 behind the choice is twofold.
 
-First, we seek draw conclusions from our model that would be applicable to
-generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
-provide this, save for a single anomaly: the value of the energy and its
-gradient at any point $z$ correlate along the $z$ direction, with
-$\overline{H_0\partial H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This
-anomalous direction must be neglected if we are to draw conclusions about
-generic functions, and the constraint surface $z^Tz=N$ is the unique surface
+First, we seek draw conclusions from our model that are applicable to generic
+holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
+this, save for a single anomaly: the value of the energy and its gradient at
+any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
+H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous direction
+must be neglected, and the constraint surface $z^Tz=N$ is the unique surface
 whose normal is parallel to $z$ and which contains the configuration space of
 the real $p$-spin model as a subspace.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
 and therefore permits easy generalization of the integration techniques
-referenced above. The same cannot be said for the space defined by $z^\dagger z=N$,
-which is topologically the $(2N-1)$-sphere and cannot admit a complex
+referenced above. The same cannot be said for the space defined by $z^\dagger
+z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
 A consequence of the constraint is that the model's configuration space is not
-compact, nor is its energy bounded. This is not necessarily a problem, as many
+compact, nor is its energy bounded. This is not necessarily problematic, as many
 related problems have similar properties but are concerned with subspaces on
-which the energy is bounded. (In fact, identifying the appropriate subspace on
-which to define one's model often requires the study of critical points in the
-whole space.) Where it might be a problem, we introduce the additional
-constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex
-manifold with boundary. We shall see that the `radius' $r$ proves an insightful
-knob in our present problem, revealing structure as it is varied. Note
-that taking $r=1$ reduces the problem to that of the ordinary $p$-spin.
-
-The critical points are of $H$ given by the solutions to the set of equations
+which the energy is bounded. (In fact, identifying the appropriate subspace
+often requires the study of critical points in the whole space.) Where it might
+become problematic, we introduce an additional constraint that bounds the
+`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$.  The resulting
+configuration space is a complex manifold with boundary. We shall see that the
+`radius' $r$ and its upper bound $R$ prove to be insightful knobs in our present
+problem, revealing structure as they are varied. Note that taking $R=1$ reduces
+the problem to that of the ordinary $p$-spin.
+
+The critical points are of $H$ given by the solutions to
 \begin{equation} \label{eq:polynomial}
   \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}}
   = p\epsilon z_i
@@ -140,7 +140,7 @@ equations of degree $p-1$, to which one must add the constraint.  In this sense
 this study also provides a complement to the work on the distribution of zeroes
 of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
 and $p\to\infty$.  We see from \eqref{eq:polynomial} that at any critical
-point $\epsilon=H/N$, the average energy.
+point $\epsilon=H_0/N$, the average energy.
 
 Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also
 one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for
@@ -149,7 +149,7 @@ function of $2N$ real variables. The number of critical points of $H$ is thus gi
 usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
-    \mathcal N&(\kappa,\epsilon,r)
+    \mathcal N&(\kappa,\epsilon,R)
       = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\hspace{6pc}\times\left|\det\begin{bmatrix}
           \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
@@ -157,11 +157,18 @@ usual Kac--Rice formula applied to $\operatorname{Re}H$:
         \end{bmatrix}\right|.
   \end{aligned}
 \end{equation}
-The Cauchy--Riemann equations may be used to write this in a manifestly complex
-way.  With the Wirtinger derivative $\partial=\frac12(\partial_x-i\partial_y)$,
-one can write $\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and
+This expression is to be averaged over $J$ to give the complexity $\Sigma$ as
+$N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica
+trick. In most of the parameter space that we shall study here, the
+\emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
+exact. 
+
+The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
+a manifestly complex way.  With the Wirtinger derivative
+$\partial\equiv\frac12(\partial_x-i\partial_y)$, one can write
+$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and
 $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
-transformations through, we have
+transformations through, one finds
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
     \mathcal N&(\kappa,\epsilon,r)
@@ -177,8 +184,9 @@ transformations through, we have
   \end{aligned}
 \end{equation}
 This gives three equivalent expressions for the determinant of the Hessian: as
-that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix,
-or the norm squared of that of an $N\times N$ complex symmetric matrix.
+that of a $2N\times 2N$ real symmetric matrix, that of the $N\times N$ Hermitian
+matrix $(\partial\partial H)^\dagger\partial\partial H$, or the norm squared of
+that of the $N\times N$ complex symmetric matrix $\partial\partial H$.
 
 These equivalences belie a deeper connection between the spectra of the
 corresponding matrices. Each positive eigenvalue of the real matrix has a
@@ -189,52 +197,13 @@ Hessian is therefore the same as the distribution of singular values of
 $\partial\partial H$, or the distribution of square-rooted eigenvalues of
 $(\partial\partial H)^\dagger\partial\partial H$.
 
-The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give
-the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N}$, a calculation
-that involves the replica trick. In most of the parameter-space that we shall
-study here, the \emph{annealed approximation} $N \Sigma \sim \log \overline{
-\mathcal N}$ is exact. 
-
 A useful property of the Gaussian $J$ is that gradient and Hessian at fixed
-$\epsilon$ are statistically independent \cite{Bray_2007_Statistics,
+energy $\epsilon$ are statistically independent \cite{Bray_2007_Statistics,
 Fyodorov_2004_Complexity}, so that the $\delta$-functions and the Hessian may
-be averaged independently. The $\delta$-functions are converted to exponentials
-by the introduction of auxiliary fields $\hat z=\hat x+i\hat y$.  The average
-of those factors over $J$ can then be performed. A generalized
-Hubbard--Stratonovich allows a change of variables from the $4N$ original
-and auxiliary fields to eight bilinears defined by $Na=z^\dagger z$, $N\hat a=\hat z^\dagger\hat z$, $N\hat c=\hat z^T\hat z$, $Nb=\hat z^\dagger z$, and $Nd=\hat z^Tz$ (and their
-conjugates). The result, to leading order in $N$, is
-\begin{equation} \label{eq:saddle}
-    \overline{\mathcal N}(\kappa,\epsilon,r)
-        = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)},
-\end{equation}
-where the argument of the exponential is
-\begin{widetext}
-  \begin{equation}
-    f=2+\frac12\log\det\frac12\begin{bmatrix}
-      1 & a & d & b \\
-      a & 1 & b^* & d^* \\
-      d & b^* & \hat c & \hat a \\
-      b & d^* & \hat a & \hat c^*
-    \end{bmatrix}
-    +\int d\lambda\,\rho(\lambda)\log|\lambda|^2
-    +p\operatorname{Re}\left\{
-      \frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b
-    \right\}.
-  \end{equation}
-  The integral of the distribution $\rho$ of eigenvalues of $\partial\partial
-  H$ comes from the Hessian and is dependant on $a$ alone. This function has an
-  extremum in $\hat a$, $b$, $\hat c$, and $d$ at which its value is
-  \begin{equation} \label{eq:free.energy.a}
-      f(a)=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-|\kappa|^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2
-      -2C_+[\operatorname{Re}(\epsilon e^{-i\theta})]^2-2C_-[\operatorname{Im}(\epsilon e^{-i\theta})]^2,
-  \end{equation}
-\end{widetext}
-where $\theta=\frac12\arg\kappa$ and
-\begin{equation}
-  C_{\pm}=\frac{a^p(1+p(a^2-1))\mp a^2|\kappa|}{a^{2p}\pm(p-1)a^p(a^2-1)|\kappa|-a^2|\kappa|^2}.
-\end{equation}
-This leaves a single parameter, $a$, which dictates the norm of $z$. 
+be averaged independently. First we shall compute the spectrum of the Hessian,
+which can in turn be used to compute the determinant. Then we will treat the
+$\delta$-functions and the resulting saddle point equations. The results of
+these calculations begin around \eqref{eq:bezout}.
 
 The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
 the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
@@ -246,28 +215,28 @@ Hessian of the unconstrained Hamiltonian is
   =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
 \end{equation}
 which makes its ensemble that of Gaussian complex symmetric matrices, when the
-direction along the constraint is neglected. Given its variances
-$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
+anomalous direction normal to the constraint surface is neglected. Given its variances
+$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and
 $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
 constant inside the ellipse
 \begin{equation} \label{eq:ellipse}
-  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
-  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
-  <\frac{p(p-1)}{2a^{p-2}}
+  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{r^{p-2}+|\kappa|}\right)^2+
+  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{r^{p-2}-|\kappa|}\right)^2
+  <\frac{p(p-1)}{2r^{p-2}}
 \end{equation}
 where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
 spectrum of $\partial\partial H$ is therefore constant inside the same ellipse
 translated so that its center lies at $-p\epsilon$.  Examples of these
 distributions are shown in the insets of Fig.~\ref{fig:spectra}.
 
-The eigenvalue spectrum of the Hessian of the real part is different from the
-spectrum $\rho(\lambda)$ of $\partial\partial H$, but rather equivalent to the
+The eigenvalue spectrum of the Hessian of the real part is not the
+spectrum $\rho(\lambda)$ of $\partial\partial H$, but instead the
 square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$;
 in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial
 H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this
 is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and
 to our knowledge a closed form is not in the literature.  We have worked out an
-implicit form for this spectrum using the replica method.
+implicit form for the singular value spectrum using the replica method.
 
 Introducing replicas to bring the partition function into the numerator of the
 Green function \cite{Livan_2018_Introduction} gives
@@ -280,25 +249,24 @@ Green function \cite{Livan_2018_Introduction} gives
       \right]
     \right\},
   \end{equation}
-  with sums taken over repeated Latin indices.  The average is then made over
+  with sums taken over repeated Latin indices. The average is then made over
   $J$ and Hubbard--Stratonovich is used to change variables to the replica matrices
-  $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^*\cdot\zeta^{(\beta)}$ and
-  $N\chi_{\alpha\beta}=\zeta^{(\alpha)}\cdot\zeta^{(\beta)}$ and a series of
-  replica vectors. The replica-symmetric ansatz leaves all off-diagonal
-  elements and vectors zero, and
-  $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
+  $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^\dagger\zeta^{(\beta)}$ and
+  $N\chi_{\alpha\beta}=(\zeta^{(\alpha)})^T\zeta^{(\beta)}$, and a series of
+  replica vectors. The replica-symmetric ansatz leaves all replica vectors
+  zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
   $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
   \begin{equation}\label{eq:green.saddle}
     \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
     \exp\left\{nN\left[
-      1+\frac{p(p-1)}{16}a^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
+      1+\frac{p(p-1)}{16}r^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
       +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
     \right]\right\}.
     \nonumber % He's too long, and we don't cite him (now)!
   \end{equation}
 \end{widetext}
 
-\begin{figure}[b]
+\begin{figure}
   \centering
 
   \includegraphics{fig/spectra_0.0.pdf}
@@ -307,25 +275,27 @@ Green function \cite{Livan_2018_Introduction} gives
   \includegraphics{fig/spectra_1.5.pdf}
 
   \caption{
-    Eigenvalue and singular value spectra of the matrix $\partial\partial H$
-    for $p=3$, $a=\frac54$, and $\kappa=\frac34e^{-i3\pi/4}$ with (a)
-    $\epsilon=0$, (b) $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
+    Eigenvalue and singular value spectra of the Hessian $\partial\partial H$
+    of the $3$-spin model with $\kappa=\frac34e^{-i3\pi/4}$. Pictured
+    distributions are for critical points at `radius' $r=\sqrt{5/4}$ and with
+    energy per spin (a) $\epsilon=0$, (b)
+    $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
     $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
-    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each inset
-    shows the support of the eigenvalue distribution. The solid line on each
-    plot shows the distribution of singular values, while the overlaid
-    histogram shows the empirical distribution from $2^{10}\times2^{10}$ complex
-    normal matrices with the same covariance and diagonal shift as
-    $\partial\partial H$.
+    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each
+    inset shows the support of the eigenvalue distribution \eqref{eq:ellipse}.
+    The solid line on each plot shows the distribution of singular values
+    \eqref{eq:spectral.density}, while the overlaid histogram shows the
+    empirical distribution from $2^{10}\times2^{10}$ complex normal matrices
+    with the same covariance and diagonal shift as $\partial\partial H$.
   } \label{fig:spectra}
 \end{figure}
 
 The argument of the exponential has several saddles. The solutions $\alpha_0$
 are the roots of a sixth-order polynomial, and the root with the smallest value
-of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct
-solution. A detailed analysis of the saddle point integration is needed to
-understand why this is so. Given such $\alpha_0$, the density of singular
-values follows from the jump across the cut, or
+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
 \begin{equation} \label{eq:spectral.density}
   \rho(\sigma)=\frac1{i\pi N}\left(
     \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma)
@@ -335,70 +305,109 @@ values follows from the jump across the cut, or
 Examples can be seen in Fig.~\ref{fig:spectra} compared with numeric
 experiments.
 
-The transition from a one-cut to two-cut singular value spectrum naturally
-corresponds to the origin leaving the support of the eigenvalue spectrum.
-Weyl's theorem requires that the product over the norm of all eigenvalues must
-not be greater than the product over all singular values \cite{Weyl_1912_Das}.
-Therefore, the absence of zero eigenvalues implies the absence of zero singular
-values. The determination of the threshold energy -- the energy at which the
-distribution of singular values becomes gapped -- is then reduced to a
-geometry problem, and yields
+The formation of a gap in the singular value spectrum naturally corresponds to
+the origin leaving the support of the eigenvalue spectrum.  Weyl's theorem
+requires that the product over the norm of all eigenvalues must not be greater
+than the product over all singular values \cite{Weyl_1912_Das}.  Therefore, the
+absence of zero eigenvalues implies the absence of zero singular values. The
+determination of the threshold energy---the energy at which the distribution
+of singular values becomes gapped---is reduced to the geometry problem of
+determining when the boundary of the ellipse defined in \eqref{eq:ellipse}
+intersects the origin, and yields
 \begin{equation} \label{eq:threshold.energy}
   |\epsilon_{\mathrm{th}}|^2
-  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2a^{p-2}}
+  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2r^{p-2}}
   {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
 \end{equation}
-for $\delta=\kappa a^{-(p-2)}$.
+for $\delta=\kappa r^{-(p-2)}$. Notice that the threshold depends on both the
+energy per spin $\epsilon$ on the `radius' $r$ of the saddle.
+
+We will now address the $\delta$-functions of \eqref{eq:complex.kac-rice}.
+These are converted to exponentials by the introduction of auxiliary fields
+$\hat z=\hat x+i\hat y$.  The average over $J$ can then be performed. A
+generalized Hubbard--Stratonovich allows a change of variables from the $4N$
+original and auxiliary fields to eight bilinears defined by $Nr=z^\dagger z$,
+$N\hat r=\hat z^\dagger\hat z$, $Na=\hat z^\dagger z$, $Nb=\hat z^Tz$, and
+$N\hat c=\hat z^T\hat z$ (and their conjugates). The result, to leading order
+in $N$, is
+\begin{equation} \label{eq:saddle}
+    \overline{\mathcal N}(\kappa,\epsilon,R)
+        = \int dr\,d\hat r\,da\,da^*db\,db^*d\hat c\,d\hat c^*e^{Nf(r,\hat r,a,b,\hat c)},
+\end{equation}
+where the argument of the exponential is
+\begin{widetext}
+  \begin{equation}
+    f=2+\frac12\log\det\frac12\begin{bmatrix}
+      1 & r & b & a \\
+      r & 1 & a^* & b^* \\
+      b & a^* & \hat c & \hat r \\
+      a & b^* & \hat r & \hat c^*
+    \end{bmatrix}
+    +\int d\lambda\,d\lambda^*\rho(\lambda)\log|\lambda|^2
+    +p\operatorname{Re}\left\{
+      \frac18\left[\hat rr^{p-1}+(p-1)|b|^2r^{p-2}+\kappa(\hat c^*+(p-1)a^2)\right]-\epsilon a
+    \right\}.
+  \end{equation}
+  The spectrum $\rho$ is given in \eqref{eq:ellipse} and is dependant on $r$ alone. This function has an
+  extremum in $\hat r$, $a$, $b$, and $\hat c$ at which its value is
+  \begin{equation} \label{eq:free.energy.a}
+      f=1+\frac12\log\left(\frac4{p^2}\frac{r^2-1}{r^{2(p-1)}-|\kappa|^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2
+      -2C_+[\operatorname{Re}(\epsilon e^{-i\theta})]^2-2C_-[\operatorname{Im}(\epsilon e^{-i\theta})]^2,
+  \end{equation}
+\end{widetext}
+where $\theta=\frac12\arg\kappa$ and
+\begin{equation}
+  C_{\pm}=\frac{r^p(1+p(r^2-1))\mp r^2|\kappa|}{r^{2p}\pm(p-1)r^p(r^2-1)|\kappa|-r^2|\kappa|^2}.
+\end{equation}
+Notice that level sets of $f$ in energy $\epsilon$ also give ellipses, but of
+different form from the ellipse in \eqref{eq:ellipse}.
 
-Given $\rho$, the integral in \eqref{eq:free.energy.a} may be preformed for
-arbitrary $a$. The resulting expression is maximized for $a=r^2$ for all
-values of $\kappa$ and $\epsilon$. Evaluating the complexity at this saddle, in
-the limit of unbounded spins, gives
+This expression is maximized for $r=R$, its value at the boundary, for
+all values of $\kappa$ and $\epsilon$. Evaluating the complexity at this
+saddle, in the limit of unbounded spins, gives
 \begin{equation} \label{eq:bezout}
-  \lim_{r\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,r)
+  \lim_{R\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,R)
   =N\log(p-1).
 \end{equation}
-This is, to this order,  precisely the Bézout bound, the maximum number of
-solutions that $N$ equations of degree $p-1$ may have
-\cite{Bezout_1779_Theorie}. That we saturate this bound is perhaps not
-surprising, since the coefficients of our polynomial equations
-\eqref{eq:polynomial} are complex and have no symmetries. Reaching Bézout in
-\eqref{eq:bezout} is not our main result, but it provides a good check.
-Analogous asymptotic scaling has been found for the number of pure Higgs states
-in supersymmetric quiver theories \cite{Manschot_2012_From}.
+This is, to leading order, precisely the Bézout bound, the maximum number of
+solutions to $N$ equations of degree $p-1$ \cite{Bezout_1779_Theorie}. That we
+saturate this bound is perhaps not surprising, since the coefficients of our
+polynomial equations \eqref{eq:polynomial} are complex and have no symmetries.
+Reaching Bézout in \eqref{eq:bezout} is not our main result, but it provides a
+good check.  Analogous asymptotic scaling has been found for the number of pure
+Higgs states in supersymmetric quiver theories \cite{Manschot_2012_From}.
 
 \begin{figure}[htpb]
   \centering
   \includegraphics{fig/complexity.pdf}
   \caption{
-    The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    the maximum `radius' $r$ at several values of $\kappa$. The dashed line
-    shows $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
+    The complexity of the 3-spin model as a function of the maximum `radius'
+    $R$ at zero energy and several values of $\kappa$. The dashed line shows
+    $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
   } \label{fig:complexity}
 \end{figure}
 
-For finite $r$, everything is analytically tractable at $\epsilon=0$, giving
+For finite $R$, everything is analytically tractable at $\epsilon=0$:
 \begin{equation} \label{eq:complexity.zero.energy}
-  \Sigma(\kappa,0,r)
-  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2r^{-4(p-1)}}{1-r^{-4}}\right).
+  \Sigma(\kappa,0,R)
+  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2R^{-4(p-1)}}{1-R^{-4}}\right).
 \end{equation}
-This is plotted as a function of $r$ for
-several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$,
-the complexity goes to negative infinity as $r\to1$, i.e., as the spins are
-restricted to the reals.  This is natural, given that the $y$ contribution to
-the volume shrinks to zero as that of an $N$-dimensional sphere $\sum_i y_i^2=N(r^2-1)$ with volume
-$\sim(r^2-1)^N$.  However, when the result is analytically continued to
+This is plotted as a function of $R$ for several values of $\kappa$ in
+Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to
+negative infinity as $R\to1$, i.e., as the spins are restricted to the reals.
+This is natural, since volume of configuration space vanishes in this limit
+like $(R^2-1)^N$. However, when the result is analytically continued to
 $\kappa=1$ (which corresponds to real $J$) something novel occurs: the
-complexity has a finite value at $r=1$. Since the $r$-dependence gives a
-cumulative count, this implies a $\delta$-function density of critical points
-along the line $y=0$.  The number of critical points contained within is
+complexity has a finite value at $R=1$. This implies a $\delta$-function
+density of critical points on the $r=1$ (or $y=0$) boundary.  The number of
+critical points contained there is
 \begin{equation}
-  \lim_{r\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,r)
+  \lim_{R\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,R)
   = \frac12N\log(p-1),
 \end{equation}
 half of \eqref{eq:bezout} and corresponding precisely to the number of critical
-points of the real $p$-spin model (note the role of conjugation symmetry,
-already underlined in \cite{Bogomolny_1992_Distribution}). The full
+points of the real $p$-spin model. (Note the role of conjugation symmetry,
+already underlined in \cite{Bogomolny_1992_Distribution}.) The full
 $\epsilon$-dependence of the real $p$-spin is recovered by this limit as
 $\epsilon$ is varied.
 
@@ -406,7 +415,7 @@ $\epsilon$ is varied.
   \centering
   \includegraphics{fig/desert.pdf}
   \caption{
-    The value of `radius' $r$ for which $\Sigma(\kappa,\epsilon,r)=0$ as a
+    The value of bounding `radius' $R$ for which $\Sigma(\kappa,\epsilon,R)=0$ as a
     function of (real) energy per spin $\epsilon$  for the 3-spin model at
     several values of $\kappa$. Above each line the complexity is positive and
     critical points proliferate, while below it the complexity is negative and
@@ -416,25 +425,25 @@ $\epsilon$ is varied.
   } \label{fig:desert}
 \end{figure}
 
-In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
-critical points are concentrated at infinite radius, i.e., at complex vectors with
-very large squared norm. For finite $N$ the average radius of critical points is likewise finite. By differentiating $\overline{\mathcal N}$ with
-respect to $r$ and normalizing, one has the distribution
-of critical points as a function of $r$. The average radius this yields is
-$\propto N^{1/4}+O(N^{-3/4})$. One therefore expects typical
-critical points to have a norm that grows modestly with system size.
+In the thermodynamic limit, \eqref{eq:complexity.zero.energy} implies that most
+critical points are concentrated at infinite radius $r$. For finite $N$ the
+average radius of critical points is likewise finite. By differentiating
+$\overline{\mathcal N}$ with respect to $R$ and normalizing, one obtains the
+distribution of critical points as a function of $r$. This yields an average
+radius proportional to $N^{1/4}$.  One therefore expects typical critical
+points to have a norm that grows modestly with system size.
 
 These qualitative features carry over to nonzero $\epsilon$. In
-Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $r$
-close to one for which there are no solutions. When $\kappa=1$---the analytic
-continuation to the real computation---the situation is more interesting. In
-the range of energies where there are real solutions this gap closes, which is
-only possible if the density of solutions diverges at $r=1$.  Another
-remarkable feature of this limit is that there is still a gap without solutions
-around `deep' real energies where there is no real solution. A moment's thought
-tells us that this is a necessity: otherwise a small perturbation of the $J$s
-could produce an unusually deep solution to the real problem, in a region where
-this should not happen.
+Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap in $r$
+close to one in which solutions are exponentially suppressed. When
+$\kappa=1$---the analytic continuation to the real computation---the situation
+is more interesting. In the range of energies where there are real solutions
+this gap closes, which is only possible if the density of solutions diverges at
+$r=1$. Outside this range, around `deep' real energies where real solutions are
+exponentially suppressed, the gap remains. A moment's thought tells us that
+this is necessary: otherwise a small perturbation of the $J$s could produce
+an unusually deep solution to the real problem, in a region where this should
+not happen.
 
 \begin{figure}[t]
   \centering
@@ -447,37 +456,38 @@ this should not happen.
   \caption{
     Energies at which states exist (green shaded region) and threshold energies
     (black solid line) for the 3-spin model with
-    $\kappa=\frac34e^{-i3\pi/4}$ and (a) $a=2$, (b) $a=1.325$, (c) $a=1.125$,
-    and (d) $a=1$. No shaded region is shown in (d) because no states exist at
+    $\kappa=\frac34e^{-i3\pi/4}$ and (a) $r=\sqrt2$, (b) $r=\sqrt{1.325}$, (c) $r=\sqrt{1.125}$,
+    and (d) $r=1$. No shaded region is shown in (d) because no states exist at
     any energy.
   } \label{fig:eggs}
 \end{figure}
 
 The relationship between the threshold and ground, or extremal, state energies
-is richer than in the real case. In Fig.~\ref{fig:eggs} these are shown in the
+is richer than in the real case.  In Fig.~\ref{fig:eggs} these are shown in the
 complex-$\epsilon$ plane for several examples. Depending on the parameters, the
-threshold might always come at smaller magnitude than the extremal state, or
-always come at larger magnitude, or cross as a function of complex argument.
-For sufficiently large $r$ the threshold always comes at larger magnitude than
-the extremal state. If this were to happen in the real case, it would likely
-imply our replica symmetric computation is unstable, since having a ground
-state above the threshold implies a ground state Hessian with many negative
-eigenvalues, a contradiction. However, this is not an obvious contradiction in
-the complex case. The relationship between the threshold, i.e., where the gap
-appears, and the dynamics of, e.g., a minimization algorithm or physical
-dynamics, are a problem we hope to address in future work.
-
- This paper provides a first step towards the study of a complex landscape with
- complex variables. The next obvious one is to study the topology of the
+threshold might have a smaller or larger magnitude than the extremal state, or
+cross as a function of complex argument.  For sufficiently large $r$ the
+threshold is always at a larger magnitude. If this were to happen in the real
+case, it would likely imply our replica symmetric computation were unstable,
+since having a ground state above the threshold implies a ground state Hessian
+with many negative eigenvalues, a contradiction. However, this is not an
+contradiction in the complex case, where the energy is not bounded from below.
+The relationship between the threshold, i.e., where the gap appears, and the
+dynamics of, e.g., a minimization algorithm, deformed integration cycle, or
+physical dynamics, are a problem we hope to address in future work.
+
+ This paper provides a first step towards the study of complex landscapes with
+ complex variables. The next obvious step is to study the topology of the
  critical points, the sets reached following  gradient descent (the
  Lefschetz thimbles), and ascent (the anti-thimbles) \cite{Witten_2010_A,
  Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward,
  Scorzato_2016_The}, which act as constant-phase integrating `contours.'
  Locating and counting the saddles that are joined by gradient lines---the
  Stokes points, which play an important role in the theory---is also well within
- reach of the present-day spin-glass literature techniques.    We anticipate
- that the threshold level, where the system develops a mid-spectrum gap, will
- play a crucial role as it does in the real case.
+ reach of the present-day spin-glass literature techniques. We anticipate
+ that the threshold level, where the system develops a mid-spectrum gap, plays
+ a crucial role in determining whether these Stokes points proliferate under
+ some continuous change of parameters.
 
 \begin{acknowledgments}
   We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions.
diff --git a/fig/complexity.pdf b/fig/complexity.pdf
index f9336bb..cd4864d 100644
Binary files a/fig/complexity.pdf and b/fig/complexity.pdf differ
diff --git a/fig/desert.pdf b/fig/desert.pdf
index 08d8f41..c9e03df 100644
Binary files a/fig/desert.pdf and b/fig/desert.pdf differ
-- 
cgit v1.2.3-54-g00ecf


From c038175a92d5773809f9d0928ba3c9f70b7056a9 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Thu, 18 Mar 2021 09:58:50 +0000
Subject: Update on Overleaf.

---
 appeal.tex |  9 +++++++--
 bezout.tex | 32 ++++++++++++++------------------
 2 files changed, 21 insertions(+), 20 deletions(-)

diff --git a/appeal.tex b/appeal.tex
index 2cf4757..04c8c96 100644
--- a/appeal.tex
+++ b/appeal.tex
@@ -52,16 +52,21 @@ referees that are beyond suspicion of incompetence or uncitedness:
 \begin{tabular}{ll}
   G Ben Arous  & Courant \\ 
   M Berry      & Bristol\\
+  Y. Fyodorov   & King's College, London \\
   Daniel Fisher& Stanford\\
-  T Lubensky   & Penn\\
+  T Lubensky   & U Penn\\
   M Moore      & Manchester \\
   E Witten     & IAS Princeton
 \end{tabular}
 
-We have also added  some very first results of the geometric implications for a
+We have also pointed out  some very first results of the geometric implications for a
 random complex landscape of our calculation, something we plan to expend on in
 a future full article.
 
+Let us conclude by remarking that, although it is probably true that this paper will not
+make more than one hundred citations next year, we are  confident that it will still 
+be considered relevant in ten years time.
+
 
 \closing{Sincerely,}
 
diff --git a/bezout.tex b/bezout.tex
index f050707..f596a89 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -69,7 +69,7 @@ $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are
 $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The
 constraint remains $z^Tz=N$.
 
-The motivations for this paper are of two types. On the practical side, there
+The motivations for this paper are of three types. On the practical side, there
 are indeed situations in which complex variables appear naturally in disordered
 problems: such is the case in which the variables are \emph{phases}, as in
 random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used
@@ -80,15 +80,14 @@ plane often uncovers underlying simplicity that is otherwise hidden, shedding
 light on the original real problem, e.g., as in the radius of convergence of a
 series.
 
-Deforming an integral in $N$ real variables to a surface of dimension $N$ in
+Finally, deforming an integral in $N$ real variables to a surface of dimension $N$ in
 $2N$-dimensional complex space has turned out to be necessary for correctly
 defining and analyzing path integrals with complex action (see
 \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the
 sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient,
 Scorzato_2016_The}.  In order to do this correctly, features of landscape
-of the action in complex space---like the relative position of its
-saddles---must be understood. Such landscapes are in general not random: here
-we propose to follow the strategy of computer science of understanding the
+of the action in complex space--- such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here
+we propose to follow standard the strategy of computer science of understanding the
 generic features of random instances, expecting that this sheds light on
 practical, nonrandom problems.
 
@@ -107,9 +106,7 @@ holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
 this, save for a single anomaly: the value of the energy and its gradient at
 any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
 H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous direction
-must be neglected, and the constraint surface $z^Tz=N$ is the unique surface
-whose normal is parallel to $z$ and which contains the configuration space of
-the real $p$-spin model as a subspace.
+thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely this.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
@@ -118,14 +115,13 @@ referenced above. The same cannot be said for the space defined by $z^\dagger
 z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
-A consequence of the constraint is that the model's configuration space is not
-compact, nor is its energy bounded. This is not necessarily problematic, as many
-related problems have similar properties but are concerned with subspaces on
-which the energy is bounded. (In fact, identifying the appropriate subspace
-often requires the study of critical points in the whole space.) Where it might
-become problematic, we introduce an additional constraint that bounds the
-`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$.  The resulting
-configuration space is a complex manifold with boundary. We shall see that the
+Imposing the constraint with  a holomorphic function
+makes the resulting configuration space is a \emph{bone fide} complex manifold, which is, as we mentioned, the
+situation we wish to model. The same cannot be said for the space defined by $z^\dagger
+z=N$, which is topologically the $(2N-1)$-sphere,  does not admit a complex
+structure, and thus yields a trivial structure of saddles.
+However, we will introduce the domains of
+`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$, as a device to classify saddles.   We shall see that the
 `radius' $r$ and its upper bound $R$ prove to be insightful knobs in our present
 problem, revealing structure as they are varied. Note that taking $R=1$ reduces
 the problem to that of the ordinary $p$-spin.
@@ -159,9 +155,9 @@ usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \end{equation}
 This expression is to be averaged over $J$ to give the complexity $\Sigma$ as
 $N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica
-trick. In most of the parameter space that we shall study here, the
+trick. Based on the experience from these problems \cite{Castellani_2005_Spin-glass},  the
 \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
-exact. 
+expected to be exact wherever the complexity is positive.
 
 The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
 a manifestly complex way.  With the Wirtinger derivative
-- 
cgit v1.2.3-54-g00ecf


From 859e78dbcd4f1b24371cbf98c247b625fcfc827b Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 18 Mar 2021 12:17:04 +0100
Subject: Small changes of Jorge's edits.

---
 bezout.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index f596a89..4b3bd19 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -86,8 +86,8 @@ defining and analyzing path integrals with complex action (see
 \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the
 sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient,
 Scorzato_2016_The}.  In order to do this correctly, features of landscape
-of the action in complex space--- such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here
-we propose to follow standard the strategy of computer science of understanding the
+of the action in complex space---such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here
+we follow the standard strategy of computer science by understanding the
 generic features of random instances, expecting that this sheds light on
 practical, nonrandom problems.
 
@@ -106,7 +106,7 @@ holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
 this, save for a single anomaly: the value of the energy and its gradient at
 any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
 H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous direction
-thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely this.
+should thus be forbidden, and the constraint surface $z^Tz=N$ accomplishes this.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
@@ -116,13 +116,13 @@ z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
 Imposing the constraint with  a holomorphic function
-makes the resulting configuration space is a \emph{bone fide} complex manifold, which is, as we mentioned, the
+makes the resulting configuration space a \emph{bone fide} complex manifold, which is, as we mentioned, the
 situation we wish to model. The same cannot be said for the space defined by $z^\dagger
-z=N$, which is topologically the $(2N-1)$-sphere,  does not admit a complex
+z=N$, which is topologically the $(2N-1)$-sphere, does not admit a complex
 structure, and thus yields a trivial structure of saddles.
-However, we will introduce the domains of
-`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$, as a device to classify saddles.   We shall see that the
-`radius' $r$ and its upper bound $R$ prove to be insightful knobs in our present
+However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq R^2$
+on the `radius' per spin as a device to classify saddles.   We shall see that this
+`radius' $r$ and its upper bound $R$ are insightful knobs in our present
 problem, revealing structure as they are varied. Note that taking $R=1$ reduces
 the problem to that of the ordinary $p$-spin.
 
@@ -155,7 +155,7 @@ usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \end{equation}
 This expression is to be averaged over $J$ to give the complexity $\Sigma$ as
 $N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica
-trick. Based on the experience from these problems \cite{Castellani_2005_Spin-glass},  the
+trick. Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},  the
 \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
 expected to be exact wherever the complexity is positive.
 
-- 
cgit v1.2.3-54-g00ecf


From 68ae488545e3acd1ff7b7a795af57c0fa10a3fc6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 18 Mar 2021 12:19:49 +0100
Subject: Small formatting change.

---
 appeal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/appeal.tex b/appeal.tex
index 04c8c96..6af3109 100644
--- a/appeal.tex
+++ b/appeal.tex
@@ -52,7 +52,7 @@ referees that are beyond suspicion of incompetence or uncitedness:
 \begin{tabular}{ll}
   G Ben Arous  & Courant \\ 
   M Berry      & Bristol\\
-  Y. Fyodorov   & King's College, London \\
+  Y Fyodorov   & King's College London \\
   Daniel Fisher& Stanford\\
   T Lubensky   & U Penn\\
   M Moore      & Manchester \\
-- 
cgit v1.2.3-54-g00ecf


From 0107fa065ab055dcd724aad1cf87c2146e31ec88 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 18 Mar 2021 15:35:18 +0100
Subject: Added referee response.

---
 referee_respose.txt | 158 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 158 insertions(+)
 create mode 100644 referee_respose.txt

diff --git a/referee_respose.txt b/referee_respose.txt
new file mode 100644
index 0000000..3b204c6
--- /dev/null
+++ b/referee_respose.txt
@@ -0,0 +1,158 @@
+----------------------------------------------------------------------
+Response to Referee A -- LZ16835/Kent-Dobias
+----------------------------------------------------------------------
+
+Referee A wrote:
+> The authors consider the mean-field p-spin spherical model with
+> *complex* variables and study the number of saddle points of the
+> energy and the eigenvalue distribution of their Hessian matrix. The
+> main result of the rather technical computation is that in a
+> particular limit (concretely kappa->1) the known results for the real
+> p-spin spherical model are reproduced, the (expected) Bézout bound for
+> the number of solutions of the saddle point equations is reached and
+> that the relationships between the “threshold” and extremal state
+> energies is richer in the complex case than in the real case.
+> 
+> I must admit that I was not able to grasp any far-reaching
+> consequences of the computational tour de force only hinted at in the
+> manuscript, and I fear that a nonexpert reader would also not be able
+> to do so. Two arguments are pushed forward by the authors to justify
+> the dissemination of their results to the broader readership of PRL:
+> One is that there are indeed situations in which complex variables
+> appear naturally in disordered system. The first example the authors
+> mention is a Hamiltonian that could be relevant for with random Laser
+> problems and was analyzed 2015 in PRA, which has up to now 30
+> citations according to Google Scholar, and the second example is a
+> Hamiltonian from sting theory that was analyzed in 2016 in JHEP, which
+> has up to now 31 citations. I do not feel that these two examples
+> prove that the enumeration of saddle points if the p-spin model is
+> important or of broad interest.
+> 
+> The second argument of the authors is that extending a real problem to
+> the complex plane often uncovers underlying simplicity that is
+> otherwise hidden, shedding light on the original real problem. Here I
+> come back to what I already mentioned above: I do not see any
+> simplicity emerging from the present calculation and I also do not see
+> the original problem in a new light. Therefore, I do not think that
+> one of the four PRL criteria is actually fulfilled and I recommend to
+> transfer the manuscript to PRE.
+
+We disagree with the referee's assessment here, as we have also explained in
+our letter to the editors. Something in particular that goes unaddressed is
+another motivation (which in the referee's defense we did not enumerate clearly
+in our draft): that understanding the distribution of complex critical points
+is necessary in the treatment of a large class of integrals involved both in
+the definition of quantum mechanics with a complex action and in ameliorating
+the sign problem in, e.g., lattice QCD.
+
+If the criteria for publication is to be "first past the post" of cited
+citations, one might examine our citations of that literature:
+
+  - Analytic continuation of Chern-Simons theory, E Witten (2011): 444 citations
+
+  - New approach to the sign problem in quantum field theories: High density
+    QCD on a Lefschetz thimble, M Cristoforetti et al (2012): 285 citations
+
+Both works are concerned with the location and relative positions of critical
+points of complex theories. In the resubmitted manuscript we have better
+emphasized this motivation.
+
+Referee A wrote:
+> Although the first part of the manuscript is well written and well
+> understandable (at least for me) from page 2 on it becomes very
+> technical and unreadable for a non-expert. If the reader skips to the
+> results and tries to understand the figures she/he is left with the
+> ubiquitous parameter a, whose physical meaning is hidden deep in the
+> saddle point calculation (“dictates the magnitude of |z|^2” – well,
+> with respect to the solutions of (3): is “a” the average value of the
+> modulus squared of the solution z’s or not?). Similar with epsilon:
+> apparently it is the average energy of the saddle point solution – why
+> not writing so also in the figure captions? The paper would profit a
+> lot from a careful rewriting of at least the result section and to
+> provide figure captions with the physical meaning of the quantities
+> and parameters shown.
+
+We thank the referee for their helpful suggestions with regards to the
+readability of our manuscript. In the resubmitted version, much has been
+rewritten for clarity. We would like to highlight several of the most
+substantive changes:
+
+  - The ubiquitous parameter 'a' was replaced by the more descriptive 'r^2', as
+    it is a sort of radius, along with a new parameter 'R^2' which bounds it.
+    Descriptions in English of these were added to the figure captions.
+
+  - The technical portion of the paper was reordered to connect better with the
+    sections preceding and following it.
+
+  - The location of the results is now indicated before the beginning of the
+    technical portion for readers interested in skipping ahead.
+
+Referee A wrote:
+> A couple of minor, technical, quibbles:
+> 
+> 1) If there is any real world application of a p-spin model with
+> complex variables it will NOT have a spherical constraint. I would
+> suggest to discuss the consequences of this constraint, which is
+> introduced for computational simplicity.
+> 
+> 2) After eq. (2): ”We choose to constrain our model by z^2=N.“ Then it
+> is not a spherical constraint any more – does it have any physical
+> relevance?
+
+We have added a more detailed discussion of the constraint to address these
+confusions, emphasizing its purpose. The new paragraphs are:
+
+> One might balk at the constraint $z^Tz=N$---which could appropriately be
+> called a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$.
+> The reasoning behind the choice is twofold.
+>
+> First, we seek draw conclusions from our model that are applicable to generic
+> holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
+> this, save for a single anomaly: the value of the energy and its gradient at
+> any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
+> H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous
+> direction should thus be forbidden, and the constraint surface $z^Tz=N$
+> accomplishes this.
+>
+> Second, taking the constraint to be the level set of a holomorphic function
+> means the resulting configuration space is a \emph{bone fide} complex
+> manifold, and therefore permits easy generalization of the integration
+> techniques referenced above. The same cannot be said for the space defined by
+> $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere and cannot admit
+> a complex structure.
+>
+> Imposing the constraint with  a holomorphic function makes the resulting
+> configuration space a \emph{bone fide} complex manifold, which is, as we
+> mentioned, the situation we wish to model. The same cannot be said for the
+> space defined by $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere,
+> does not admit a complex structure, and thus yields a trivial structure of
+> saddles.  However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq
+> R^2$ on the `radius' per spin as a device to classify saddles.   We shall see
+> that this `radius' $r$ and its upper bound $R$ are insightful knobs in our
+> present problem, revealing structure as they are varied. Note that taking
+> $R=1$ reduces the problem to that of the ordinary $p$-spin.
+
+Referee A wrote:
+> 3) On p.2: “…a, which dictates the magnitude of |z|^2, or
+> alternatively the magnitude y^2 of the imaginary part. The last part
+> is hard to understand, should be explained.
+
+We thank the referee for pointing out this confusing statement, which was
+unnecessary and removed.
+
+> 4) On p.2: “In most the parameter space we shall study her, the
+> annealed approximation is exact.” I think it is necessary to provide
+> some evidence her, because the annealed approximation is usually a
+> pretty severe approximation.
+
+We have nuanced the statement in question and added a citation to a review
+article which outlines the reasoning for analogous models. The amended sentence
+reads:
+
+> Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},
+> the \emph{annealed approximation} $N\Sigma\sim\log\overline{\mathcal N}$ is
+> expected to be exact wherever the complexity is positive.
+
+Sincerely,
+Jaron Kent-Dobias & Jorge Kurchan
+
-- 
cgit v1.2.3-54-g00ecf