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Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 8 |
1 files changed, 6 insertions, 2 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 30d450e..3b06750 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -314,7 +314,7 @@ This will turn out to be important when we discriminate between counting all sol \end{aligned} \end{equation} -the question of independence \cite{Bray_2007_Statistics} + \begin{equation} \begin{aligned} @@ -323,7 +323,11 @@ the question of independence \cite{Bray_2007_Statistics} \times \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|} \end{aligned} -\end{equation}{\bf The average over disorder breaks into a product of two independent averages, one for the gradient factor and one for the determinant. The integration of all variables, including the disorder in the last factor, may be restricted to the domain such that the matrix $\partial\partial H(s_a)-\mu I$ has a specified number of negative eigenvalues. Fyodorov? } +\end{equation}{\bf +As noted by Bray and Dean \cite{Bray_2007_Statistics}, gradient and Hessian are independent +for a Gaussian distribution, and +the average over disorder breaks into a product of two independent averages, one for the gradient factor and one for the determinant. The integration of all variables, including the disorder in the last factor, may be restricted to the domain such that the matrix $\partial\partial H(s_a)-\mu I$ has a specified number of negative eigenvalues (the index {\cal{I}} of the saddle), +(see Fyodorov \cite{fyodorov2007replica} for a detailed discussion) } \begin{equation} \begin{aligned} |