Many small changes.

1 files changed, 74 insertions, 71 deletions

diff --git a/bezout.tex b/bezout.tex
index 61aa55e..67a764b 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -42,40 +42,42 @@
 \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
+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:
+defined by the energy
 \begin{equation} \label{eq:bare.hamiltonian}
   H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
-where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are
-real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single  term
-of a given $p$, this is known as the `pure $p$-spin' model, the case we shall
-study here.  This problem has been studied also  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
+where $J$ is a symmetric tensor whose elements are real Gaussian variables and
+$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. If there is a
+single  term of a given $p$, this is known as the `pure $p$-spin' model, the
+case we shall study here. 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
-high-energy configuration \cite{Cugliandolo_1993_Analytical}.  Thanks to the
-relative simplicity of the energy, all these approaches are possible
-analytically in the large $N$ limit.
+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 shall extend the study to the case  where the variables are complex
-$z\in\mathbb C^N$ and $J$ is a symmetric tensor whose elements are 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 becomes $z^2=N$.
+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
+$\overline{J^2}=\kappa\overline{|J|^2}$ 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
+very close to ours has been proposed is the quiver Hamiltonians
+\cite{Anninos_2016_Disordered} modeling black hole horizons in the
 zero-temperature limit.  
 
 There is however a more fundamental reason for this study: we know from
@@ -92,33 +94,33 @@ 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.
 
-
-
-
-  For our model the  constraint we  choose  $z^2=N$,
-rather than $|z|^2=N$, in order to preserve the holomorphic nature of the
-functions. In addition, the  nonholomorphic spherical constraint has a
-disturbing lack of critical points nearly everywhere, since $0=\partial^*
-H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint.  It is enforced using the method of Lagrange multipliers:
-introducing the $\epsilon\in\mathbb C$, this gives
+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 easy to see that {\em for a pure $p$-spin}, at  any critical point
-$\epsilon=H/N$, the average energy.
-
-Critical points are given by the set of equations:
-
+For a \emph{pure} $p$-spin, $\epsilon=H/N$ -- 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
 \begin{equation}
-\frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i, i_2\cdots i_p}z_{i_2}\cdots z_{i_p} = \epsilon z_i
+  \frac{p}{p!}\sum_{j_2\cdots j_p}^NJ_{ij_2\cdots j_p}z_{j_2}\cdots z_{j_p} = \epsilon z_i
 \end{equation}
-which for given $\epsilon$ are a set of $N$ equations of degree $p-1$, to which one must add the constraint condition.
-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 $p \rightarrow \infty$.
-
-Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical  point of $\Im H_0$. 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 $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
+for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ are 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 $p \rightarrow \infty$.
+
+Since $H$ is holomorphic, a 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.
 
 Writing $z=x+iy$, $\operatorname{Re}H$ can be
 interpreted as a real function of $2N$ real variables. The number of critical
@@ -135,7 +137,7 @@ points it has is given by the usual Kac--Rice formula:
 \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
-$\partial=\partial_x-i\partial_y$, one can write
+$\partial=\frac12(\partial_x-i\partial_y)$, 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
@@ -158,14 +160,13 @@ 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.
 
 These equivalences belie a deeper connection between the spectra of the
-corresponding matrices: each 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
+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 $(\partial\partial
-H)^\dagger\partial\partial H$.
+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
 $N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation
@@ -210,22 +211,24 @@ where
     \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
+    \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
   $\kappa\in\mathbb R$)
   \begin{equation} \label{eq:free.energy.a}
-    \begin{aligned}
-      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 \\
-          &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Re}\epsilon)^2
-          -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Im}\epsilon)^2,
-    \end{aligned}
+      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,
   \end{equation}
 \end{widetext}
-This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot
-z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter
-vanishes as $a\to1$, where we should recover known results for the real
+where
+\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},
+\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 $\partial\partial
@@ -233,28 +236,28 @@ H=\partial\partial H_0-p\epsilon I$, 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
-\eqref{eq:bare.hamiltonian} is
+shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. 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}
-{\bf \color{red} restricting to  directions proportional to $z$, i.e. orthogonal to the constraint},  these makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
-$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
-$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
-constant inside the ellipse
+which makes its ensemble that of Gaussian complex symmetric matrices. 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
 \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 $\partial\partial H$ therefore is that of an ellipse in the complex
-plane whose center lies at $-p\epsilon$. Examples of these distributions are
-shown in the insets of Fig.~\ref{fig:spectra}.
+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
+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