From 4b623a36321f1f8225434be55d89fb5ac94b5946 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 13:49:15 +0100
Subject: Added introduction to complex Kac-Rice.

---
 bezout.tex | 30 ++++++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)

diff --git a/bezout.tex b/bezout.tex
index 53cc5e9..f877d05 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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}
 
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