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-rw-r--r--bezout.tex6
1 files changed, 5 insertions, 1 deletions
diff --git a/bezout.tex b/bezout.tex
index 42d4b14..58ea9cc 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -84,13 +84,17 @@ $\mathop{\mathrm{Re}}H$. 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)
+ \mathcal N_J(\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}
+This expression is to be averaged over the $J$'s as
+$\Sigma=
+\overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In
+
The Cauchy--Riemann relations imply that the matrix is of the form:
\begin{equation} \label{eq:real.kac-rice1}
\begin{bmatrix}