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\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
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\begin{document}

\title{Complex complex landscapes: 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}
  The complexity of the complex $p$-spin model saturates the Bézout bound \cite{Bezout_1779_Theorie}.
\end{abstract}

\maketitle

\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 the $z$ are constrained by $z\cdot z=N$ 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.

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(\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 the \eqref{eq:real.kac-rice} in the manifestly complex
form
\begin{equation} \label{eq:complex.kac-rice}
  \mathcal N(\epsilon)
    = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
      |\det\partial\partial H|^2.
\end{equation}

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