diff options
| -rw-r--r-- | bezout.tex | 6 |
1 files changed, 5 insertions, 1 deletions
@@ -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} |