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| -rw-r--r-- | bezout.tex | 19 |
1 files changed, 10 insertions, 9 deletions
@@ -131,15 +131,16 @@ singular values of $\partial\partial H$, while both are the same as the distribution of square-rooted eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$. -{\color{red} {\bf perhaps not here} This expression \eqref{eq:complex.kac-rice} 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 property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{Bray_2007_Statistics, Fyodorov_2004_Complexity}, -so that we may treat the delta-functions and the Hessians as independent. - -} +The expression \eqref{eq:complex.kac-rice} 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 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 property of the Gaussian distributions is that gradient and Hessian +may be seen to be independent \cite{Bray_2007_Statistics, +Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the +Hessians as independent. We compute each by taking the saddle point. The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial H=\partial\partial H_0-p\epsilon I$, or the Hessian of |