From e173cb736bc07937d61564f303eab5d0b60f62ed Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Mon, 7 Dec 2020 16:00:35 +0000 Subject: Update on Overleaf. --- bezout.tex | 11 ++++++++--- 1 file 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} -- cgit v1.2.3-70-g09d2