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\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, yet very rich model 
\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 $z\in\mathbb C^N$ is constrained by $z^2=N$ and $J$ is a symmetric tensor
whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and
$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter
$|\kappa|<1$. The constraint 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}
At any critical point $\epsilon=H/N$, the average energy.

When compared with $z^*z=N$, the constraint $z^2=N$ may seem an unnatural
extension of the real $p$-spin spherical model. However, a model with this
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.

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}
  \mathcal N(\kappa,\epsilon)
    = \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H)
      \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{equation}
The Cauchy--Riemann relations can be used to simplify this. 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(\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 $\partial_i\partial_j
H=\partial_i\partial_j H_0-p\epsilon\delta_{ij}$, 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}
which makes its ensemble that of Gaussian complex symmetric matrices, whose
spectrum is constant inside the support of a certain ellipse and zero
everywhere else \cite{Nguyen_2014_The}. Given its variances
$\langle|\partial_i\partial_j H_0|^2\rangle=p(p-1)a^{p-2}/2N$ and
$\langle(\partial_i\partial_j H_0)^2\rangle=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
\begin{equation} \label{eq:ellipse}
  \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{1+|\kappa|/a^{p-2}}\right)^2+
  \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{1-|\kappa|/a^{p-2}}\right)^2
  <\frac12p(p-1)a^{p-2}
\end{equation}
where $\theta=\frac12\arg\kappa$.

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