Many more small changes.

2 files changed, 206 insertions, 217 deletions

diff --git a/bezout.bib b/bezout.bib
index 630c59e..a3c937e 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -175,6 +175,20 @@
  doi = {10.1103/physrevlett.71.173}
 }
 
+@article{Dyson_1962_A,
+ author = {Dyson, Freeman J.},
+ title = {A Brownian-Motion Model for the Eigenvalues of a Random Matrix},
+ journal = {Journal of Mathematical Physics},
+ publisher = {AIP Publishing},
+ year = {1962},
+ month = {11},
+ number = {6},
+ volume = {3},
+ pages = {1191--1198},
+ url = {https://doi.org/10.1063%2F1.1703862},
+ doi = {10.1063/1.1703862}
+}
+
 @article{Fyodorov_2004_Complexity,
  author = {Fyodorov, Yan V.},
  title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices},
diff --git a/bezout.tex b/bezout.tex
index 8c24d59..5f55300 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -29,100 +29,98 @@
   of degree $p-1$.  We solve for $\overline{\mathcal{N}}$, the number of
   solutions averaged over randomness in the $N\to\infty$ limit.  We find that
   it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$.
-  The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b
-  B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$ are GOE matrices and $a-d$
-  real.  Its spectrum has a transition from one-cut to two-cut that generalizes
-  the notion  of `threshold level' that is well-known in the real problem.  The
-  results from the real problem are recovered in the limit of real disorder. In
-  this case, only the square-root of the total number solutions are real.  In
-  terms of real and imaginary parts of the energy, the solutions are divided in
-  sectors where the saddles have different topological properties.
+  The Hessian of each saddle is given by a random matrix of the form $C^\dagger
+  C$, where $C$ is a complex Gaussian matrix with a shift to its diagonal.  Its
+  spectrum has a transition where a gap develops that generalizes the notion of
+  `threshold level' well-known in the real problem.  The results from the real
+  problem are recovered in the limit of real parameters. In this case, only the
+  square-root of the total number of solutions are real.  In terms of the
+  complex energy, the solutions are divided into sectors where the saddles have
+  different topological properties.
 \end{abstract}
 
 \maketitle
 
-Spin-glasses have long been considered the paradigm of `complex landscapes' of
-many variables, a subject that includes neural networks and optimization
-problems, most notably constraint satisfaction ones.  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 have long been considered the paradigm of many variable `complex
+landscapes,' a subject that includes neural networks and optimization problems,
+most notably constraint satisfaction.  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}
 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 studied in the algebra
-\cite{Cartwright_2013_The} and probability literature
+$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. 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
 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
+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.
+large-$N$ limit.
 
-In this paper we extend the study to the case where the variables are complex:
-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
+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 $z^2=N$.
 
 The motivations for this paper are of two types. On the practical side, there
-are indeed situations in which  complex variables  in a disorder problem appear
-naturally: such is the case in which they are {\em phases}, as in random laser
-problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian
-very close to ours has been proposed is the quiver Hamiltonians
-\cite{Anninos_2016_Disordered} modeling black hole horizons in the
-zero-temperature limit.  
+are indeed situations in which complex variables appear naturally in disordered
+problems: such is the case in which they are \emph{phases}, as in 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}.  
 
 There is however a more fundamental reason for this study: we know from
-experience that extending a problem to the complex plane often uncovers an
-underlying simplicity that is hidden in the purely real case. Consider, for
-example, the procedure of starting from a simple, known Hamiltonian $H_{00}$
-and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from
-$\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$
-is a polynomial of degree $p$  chosen to have simple, known saddles. Because we
-are working in complex variables, and the saddles are simple all the way (we
-shall confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, while  with
-real variables minima of functions appear and disappear, and this procedure is
-not possible. The same idea may be implemented by performing diffusion in the
-$J$'s, and following the roots, in complete analogy with Dyson's stochastic
-dynamics.
+experience that extending a real problem to the complex plane often uncovers
+underlying simplicity that is otherwise hidden. Consider, for example, the
+procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying
+$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$
+to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a
+polynomial of degree $p$  chosen to have simple, known saddles. Because we are
+working in complex variables, and the saddles are simple all the way (we shall
+confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$,
+while  with real variables minima of functions appear and disappear, and this
+procedure is not possible. The same idea may be implemented by performing
+diffusion in the $J$s and following the roots, in complete analogy with Dyson's
+stochastic dynamics \cite{Dyson_1962_A}.
 
 The spherical constraint is enforced using the method 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}
-It is easily shown that $\epsilon=H/N$ -- the average energy -- at any
-critical point. We choose to constrain our model by $z^2=N$ rather than
-$|z|^2=N$ in order to preserve the holomorphic nature of $H$. In addition, the
-nonholomorphic spherical constraint has a disturbing lack of critical points
-nearly everywhere: if $H$ was so constrained, then $0=\partial^* H=-p\epsilon
-z$ would only be satisfied for $\epsilon=0$.  
-
-The critical points are given by the solutions to the set of equations
+At any critical point, $\epsilon=H/N$, the average energy. 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 $\epsilon=0$.  
+
+The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation}
-  \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}} = \epsilon z_i
+  \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
 \end{equation}
-for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ is a set of $N$
+for all $i=\{1,\ldots,N\}$, which for fixed $\epsilon$ is a set of $N$
 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 $p \rightarrow \infty$.
+and $p\to\infty$.
 
-Since $H$ is holomorphic, a critical point of $\operatorname{Re}H$ is also a
+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 number of critical points of $\operatorname{Re}H$. From each
-critical point emerges a gradient line of $\operatorname{Re}H$, which is also
-one of constant $\operatorname{Im}H$ and therefore constant phase.
+therefore the same as that of $\operatorname{Re}H$. From each critical point
+emerges a gradient line of $\operatorname{Re}H$, which is also one of constant
+$\operatorname{Im}H$ and therefore constant phase.
 
-Writing $z=x+iy$, $\operatorname{Re}H$ can be
-interpreted as a real function of $2N$ real variables. The number of critical
-points it has is given by the usual Kac--Rice formula:
+Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function
+of $2N$ real variables. Its number of critical points is given by the usual
+Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
     \mathcal N_J(\kappa,\epsilon)
@@ -133,12 +131,11 @@ points it has is given by the usual Kac--Rice formula:
         \end{bmatrix}\right|.
   \end{aligned}
 \end{equation}
-The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
-a manifestly complex way.  Using the Wirtinger derivative
-$\partial=\frac12(\partial_x-i\partial_y)$, 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
+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
+$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
+transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
     &\mathcal N_J(\kappa,\epsilon)
@@ -159,46 +156,35 @@ 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
-negative partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is
-an eigenvalue of the Hermitian matrix and $|\lambda|$ is a \emph{singular
-value} of the complex symmetric matrix. The distribution of positive
-eigenvalues of the Hessian is therefore the same as the distribution of
-singular values of $\partial\partial H$,  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 the $J$'s as
-$N \Sigma= \overline{\log\mathcal N} = \int dJ \, \log \mathcal N_J$, a calculation
-that involves the replica trick. In most the parameter-space that we shall
-study here, the {\em annealed approximation} $N \Sigma \sim \log \overline{
-\mathcal N} = \log\int dJ \, \mathcal N_J$ is exact. 
-
-A useful property of the Gaussian distributions is that gradient and Hessian
-for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics,
-Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the
-Hessians as independent. We compute each by taking the saddle point. The
-$\delta$-functions 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 then allows a change of
-variables from the $4N$ original and auxiliary fields to eight bilinears
-defined by
-\begin{equation}
-  \begin{aligned}
-    Na=z^*\cdot z
-    &&
-    N\hat c=\hat z\cdot\hat z
-    &&
-    Nb=\hat z^*\cdot z \\
-    N\hat a=\hat z^*\cdot\hat z
-    &&
-    Nd=\hat z\cdot z
-  \end{aligned}
-\end{equation}
-and their conjugates. The result is, to leading order in $N$,
+negative partner. For each such pair $\pm\lambda$, $\lambda^2$ is an eigenvalue
+of the Hermitian matrix and $|\lambda|$ is a \emph{singular value} of the
+complex symmetric matrix. The distribution of positive eigenvalues of the
+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 $N \Sigma= \overline{\log\mathcal N} = \int dJ \,
+\log \mathcal N_J$, a calculation that involves the replica trick. In most 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. 
+
+A useful property of the Gaussian $J$ is that gradient and Hessian at fixed
+$\epsilon$ 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 $Na=|z|^2$, $N\hat a=|\hat
+z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their
+conjugates). The result, to leading order in $N$, is
 \begin{equation} \label{eq:saddle}
-    \overline{\mathcal N_J}(\kappa,\epsilon)
+    \overline{\mathcal N}(\kappa,\epsilon)
         = \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
+where the argument of the exponential is
 \begin{widetext}
   \begin{equation}
     f=2+\frac12\log\det\frac12\begin{bmatrix}
@@ -207,64 +193,61 @@ where
       d & b^* & \hat c & \hat a \\
       b & d^* & \hat a & \hat c^*
     \end{bmatrix}
-    +\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\}
     +\int d\lambda\,\rho(\lambda)\log|\lambda|^2
+    +\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\}.
     \nonumber % He's too big!
   \end{equation}
-  where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of
-  $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in
-  $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with
-  $\kappa\in\mathbb R$)
+  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 a
+  maximum 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
-      -C_+(a)(\operatorname{Re}\epsilon)^2-C_-(a)(\operatorname{Im}\epsilon)^2,
+      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
+where $\theta=\frac12\arg\kappa$ and
 \begin{equation}
-  C_{\pm}(a)=\frac{a^p(1+p(a^2-1))\mp a^2\kappa}{a^{2p}\pm a^p(a^2-1)(p-1)-a^2\kappa^2},
+  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.
 
-The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
-H=\partial\partial H_0-p\epsilon I$, or the Hessian of
-\eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The
-eigenvalue distribution $\rho$ of the constrained Hessian is therefore related
-to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar
-shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of the unconstrained Hamiltonian is
+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
+distribution $\rho$ is therefore related to the unconstrained distribution
+$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The
+Hessian of the unconstrained Hamiltonian is
 \begin{equation} \label{eq:bare.hessian}
   \partial_i\partial_jH_0
   =\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
+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
-$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, its distribution of
-eigenvalues $\rho_0(\lambda)$ is constant inside the ellipse
+$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ 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}}
 \end{equation}
 where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
-spectrum of $\partial\partial H$ -- the constrained Hessian -- is therefore
-that of the same ellipse whose 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 the one we need for
-our Kac--Rice formula. It is  different from the spectrum $\partial\partial H$,
-but rather equivalent  to the square-root eigenvalue spectrum of
-$(\partial\partial H)^\dagger\partial\partial H$, in other words, the singular
-value spectrum of $\partial\partial H$. When $\kappa=0$ and the elements of $J$
-are standard complex normal, this corresponds to 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 saddle point of a replica symmetric calculation for the
-Green function.
+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
+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.
 
 \begin{figure}[htpb]
   \centering
@@ -288,8 +271,8 @@ Green function.
   } \label{fig:spectra}
 \end{figure}
 
-Introducing replicas to bring the partition function to
-the numerator of the Green function \cite{Livan_2018_Introduction} gives
+Introducing replicas to bring the partition function into the numerator of the
+Green function \cite{Livan_2018_Introduction} gives
 \begin{widetext}
   \begin{equation} \label{eq:green.replicas}
     G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
@@ -297,15 +280,15 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
       \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
         -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
       \right]
-    \right\}
+    \right\},
   \end{equation}
-  with sums taken over repeated latin indices.
-  The average can then be made over $J$ and Hubbard--Stratonovich used to change
-  variables to replica matrices
+  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. Taking the replica-symmetric ansatz leaves all off-diagonal elements
-  and vectors zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
+  $N\chi_{\alpha\beta}=\zeta^{(\alpha)}\cdot\zeta^{(\beta)}$ and a series of
+  replica vectors. The replica-symmetric ansatz leaves all off-diagonal
+  elements and vectors zero, and
+  $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
   $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. 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
@@ -317,8 +300,8 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
   \end{equation}
 \end{widetext}
 The argument of the exponential has several saddles. The solutions $\alpha_0$
-are the roots of a sixth-order polynomial, but the root with the
-smallest value of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct
+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
@@ -335,7 +318,7 @@ 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 therefore reduced to a
+distribution of singular values becomes gapped -- is then reduced to a
 geometry problem, and yields
 \begin{equation} \label{eq:threshold.energy}
   |\epsilon_{\mathrm{th}}|^2
@@ -344,79 +327,59 @@ geometry problem, and yields
 \end{equation}
 for $\delta=\kappa a^{-(p-2)}$.
 
-With knowledge of this distribution, the integral in \eqref{eq:free.energy.a}
-may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$,
-the resulting expression is always maximized for $a=\infty$. Taking this saddle
-gives
+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
 \begin{equation} \label{eq:bezout}
   \log\overline{\mathcal N}(\kappa,\epsilon)
-  ={N\log(p-1)}
+  =N\log(p-1).
 \end{equation}
 This is precisely the Bézout bound, the maximum number of solutions that $N$
 equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is
-gained by looking at the count as a function of $a$, defined by
-\begin{equation} \label{eq:count.def.marginal}
-  {\mathcal N}(\kappa,\epsilon,a)
-  ={\mathcal N}(\kappa,\epsilon/ \sum_i y_i^2<Na)
-\end{equation}
-and likewise the $a$-dependant complexity
-$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}_a(\kappa,\epsilon,a)$
-the large-$N$ limit, the $a$-dependant expression may be considered the
-cumulative number of critical points up to the value $a$.
-
-The integral in \eqref{eq:free.energy.a} can only be performed explicitly for
-certain ellipse geometries. One of these is at $\epsilon=0$ any values of
-$\kappa$ and $a$, which yields the $a$-dependent complexity
+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 $\log(p-1)$.
+  } \label{fig:complexity}
+\end{figure}
+
+Everything is analytically tractable for $\epsilon=0$, 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).
 \end{equation}
 Notice that the limit of this expression as $a\to\infty$ corresponds with
-\eqref{eq:bezout}, as expected. Equation \eqref{eq:complexity.zero.energy} is 
-plotted as a function of $a$ 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 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 $\sim(a-1)^N$.  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 cumulative count, this implies a $\delta$-function
-density of critical points along the line $y=0$.  The number of critical points
-contained within is
+\eqref{eq:bezout}, as expected. This is plotted as a function of $a$ 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
+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 with volume
+$\sim(a-1)^N$.  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
+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)
   = \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.
 
 \begin{figure}[htpb]
   \centering
-  \includegraphics{fig/complexity.pdf}
-  \caption{
-    The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    $a$ 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}
-
-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. For the case $\kappa=1$ -- the
-analytic continuation to the usual real computation -- the situation is more
-interesting. In the range of energies where there are real solutions this gap
-closes, and this is only possible if the density of solutions diverges at
-$a=1$.  Another remarkable feature of the limit $\kappa=1$ 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 a real, unusually deep solution
-for the real problem, in a region where we expect this not to happen.
-
-\begin{figure}[htpb]
-  \centering
   \includegraphics{fig/desert.pdf}
   \caption{
     The minimum value of $a$ for which the complexity is positive as a function
@@ -425,20 +388,17 @@ for the real problem, in a region where we expect this not to happen.
   } \label{fig:desert}
 \end{figure}
 
-The relationship between the threshold and ground -- or more generally, extremal -- state energies 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 line always come at smaller magnitude than the ground state, or always
-come at larger magnitude than the ground state, or cross as a
-function of complex argument. For sufficiently large $a$ the threshold always
-comes at larger magnitude than the ground state. If this were to happen in the
-real case, it would likely imply our replica symmetric computation is unstable,
-as having the ground state above the threshold would imply a ground state
-Hessian with many negative eigenvalues, a contradiction with the notion of a
-ground state. 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.
+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
+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
+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.
 
 \begin{figure}[htpb]
   \centering
@@ -452,15 +412,30 @@ physical dynamics, are a problem we hope to address in future work.
     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 an
+    and (d) $a=1$. No shaded region is shown in (d) because no states exist at
     any energy.
   } \label{fig:eggs}
 \end{figure}
 
-This paper provides a first step for the study of a complex landscape with complex variables. The next obvious one
-is to study the topology of the critical points and the lines of constant phase.
-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.
+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
+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
+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 for the study of a complex landscape with
+complex variables. The next obvious one is to study the topology of the
+critical points and the lines of constant phase.  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.
 
 \begin{acknowledgments}
   JK-D and JK are supported by the Simons Foundation Grant No.~454943.