\documentclass[aps,prl,reprint,longbibliography,floatfix,fleqn]{revtex4-2}

\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
\usepackage[T1]{fontenc} % vector fonts plz
\usepackage[
  colorlinks=true,
  urlcolor=purple,
  citecolor=purple,
  filecolor=purple,
  linkcolor=purple
]{hyperref} % ref and cite links with pretty colors
\usepackage{amsmath, amssymb, graphicx, xcolor} % standard packages

\begin{document}

\title{Complex complex landscapes: I: saturating the Bézout bound} % change me

\author{Jaron Kent-Dobias}
\author{Jorge Kurchan}

\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}

\date\today

\begin{abstract}
  We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which all its $N$ variables are allowed to be complex. The problem becomes
  a system of $N$ random equations of degree $p-1$. 
  We solve for quantities averaged over randomness in the $N \rightarrow \infty$ limit. 
  We show that the number of solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1) $\cite{Bezout_1779_Theorie}.
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.
In the case that the disorder is itself real, 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.
\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 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 attacked from several angles: the replica trick to
compute the Boltzmann--Gibbs distribution, 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, and
the gradient-descent -- or more generally Langevin -- dynamics staring from a
high-energy configuration.  Thanks to the relative simplicity of the energy,
all these approaches are possible analytically in the large $N$ limit.

In this paper we shall extend the study to the case  where $z\in\mathbb C^N$ are  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 $z^2=N$.

The motivations for this paper are of two types. On the practical side, there are situations in which  complex variables
have 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, etc}. 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.  

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 roots. Because we are working in
complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. 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.


Let us go back to our model.
For the  constraint we  choose here $z^2=N$, 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
\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.


Since $H$ is holomorphic, a point is a critical point of its real part if and
only if it is also a critical point of its imaginary part. The number of
critical points of $H$ is therefore the number of critical points of
$\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ 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:
\begin{equation} \label{eq:real.kac-rice}
  \begin{aligned}
    \mathcal N_J(\kappa,\epsilon)
      &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\
      &\qquad\times\left|\det\begin{bmatrix}
          \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\
          \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
        \end{bmatrix}\right|.
  \end{aligned}
\end{equation}
{\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as 
$N \Sigma= 
\overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
\ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact.

A useful property of the Gaussian distributions is that gradient and Hessian 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.

}

The Cauchy--Riemann relations imply that the matrix is of the form:
\begin{equation} \label{eq:real.kac-rice1}
      \begin{bmatrix}
        \bar A & \bar B \\
        \bar B & -\bar A
      \end{bmatrix}
\end{equation}
with $\bar A=-\mathop{\mathrm{Re}}\partial\partial H$ and $\bar B=-\mathop{\mathrm{Im}}\partial\partial H$ Gaussian real symmetric matrices, \emph{correlated}, as we shall see..
Using the Wirtinger
derivative $\partial=\partial_x-i\partial_y$, one can write
$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. With similar
transformations, the eigenvalue spectrum of the Hessian of
$\mathop{\mathrm{Re}}H$ can be shown to be equivalent to the singular value
spectrum of the Hessian $\partial\partial H$ of $H$, and as a result the
determinant that appears above is equivalent to $|\det\partial\partial H|^2$.
This allows us to write \eqref{eq:real.kac-rice} in the manifestly complex
form
\begin{equation} \label{eq:complex.kac-rice}
  \mathcal N_J(\kappa,\epsilon)
    = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
      |\det\partial\partial H|^2.
\end{equation}

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
\eqref{eq:bare.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}

{\color{red} \bf here I would explain the question of the det and also of the appearance of the gap, would draw a picture of ellipse etc, and would send the reader to an appendix for most of this part of the calculation}

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
$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
\begin{equation} \label{eq:ellipse}
  \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
  \left(\frac{\mathop{\mathrm{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
s0pectrum of $\partial\partial H$ therefore is that of an ellipse whose center
is shifted by $p\epsilon$.

The eigenvalue spectrum of the Hessian of the real part, or equivalently the
eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is 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 known.  We have worked out an implicit form for
this spectrum using the saddle point of a replica calculation for the Green
function. blah blah blah\dots

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.

% This is kind of a boring definition...
\begin{equation} \label{eq:count.def.marginal}
  \overline{\mathcal N}(\kappa,\epsilon)
  =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
\end{equation}

\begin{equation} \label{eq:count.zero.energy}
  \overline{\mathcal N}(\kappa,0,a)
  =\left[(p-1)a^{p-1}\sqrt{\frac{1-a^{-2}}{a^{2(p-1)}-|\kappa|^2}}\right]^N
\end{equation}

\begin{equation}
  \overline{\mathcal N}(\kappa,\epsilon)
  =\lim_{a\to\infty}\overline{\mathcal N}(\kappa,\epsilon,a)
  =(p-1)^N
\end{equation}

For $|\kappa|<1$,
\begin{equation}
  \lim_{a\to1}\overline{\mathcal N}(\kappa,\epsilon,a)
  =0
\end{equation}

\begin{equation}
  \lim_{a\to1}\overline{\mathcal N}(1,0,a)
  =(p-1)^{N/2}
\end{equation}

\begin{equation} \label{eq:threshold.energy}
  |\epsilon_{\mathrm{th}}|^2
  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2a^{p-2}}
  {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
\end{equation}
for $\delta=\kappa a^{-(p-2)}$.


{\color{teal} {\bf somewhere else}

Another instrument we have to study this problem is to compute the following partition function:

\begin{equation}
    Z= \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0}
    \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right) 
\end{equation}
The energy $\Re H_0, \Im H_0$ are in a one-to one relation with the temperatures $\beta_R,\beta_I$. The entropy $S(a,H_0) = \ln Z+ +\beta_{R} \langle \Re H_0 \rangle +\beta_I \langle \Im H_0\rangle$
is the logarithm of the number of configurations of a given 

}
\bibliographystyle{apsrev4-2}
\bibliography{bezout}

\end{document}
