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authorJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-07 17:00:38 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-07 17:00:38 +0100
commitfbb4ec482ce847c6590a98876e9df63a4e3a8283 (patch)
treea4cf1631e7940e9a4cc0aa5517cb3ae3c659280f
parentbc8dc51857c5e4bb8f9722bd476e8a18436f20a6 (diff)
parente173cb736bc07937d61564f303eab5d0b60f62ed (diff)
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Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484
-rw-r--r--bezout.tex11
1 files changed, 8 insertions, 3 deletions
diff --git a/bezout.tex b/bezout.tex
index 3d58342..36434e0 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
+{\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}