Update on Overleaf.

1 files changed, 8 insertions, 3 deletions

diff --git a/bezout.tex b/bezout.tex
index 2c4dbc5..81fe628 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -91,9 +91,14 @@ points it has is given by the usual Kac--Rice formula:
         \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 most of 
+{\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as 
+$N \Sigma= 
+\overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
+\ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact.
+
+A useful propert
+
+}
 
 The Cauchy--Riemann relations imply that the matrix is of the form:
 \begin{equation} \label{eq:real.kac-rice1}