Update on Overleaf.

1 files changed, 5 insertions, 1 deletions

diff --git a/bezout.tex b/bezout.tex
index 87b0a9f..e688200 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}