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@@ -41,6 +41,36 @@ multipliers: introducing the $\epsilon\in\mathbb C$, this gives \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} + \bibliographystyle{apsrev4-2} \bibliography{bezout} |