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| -rw-r--r-- | frsb_kac-rice.tex | 21 |
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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 9047178..ec71085 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -404,18 +404,19 @@ where the function $\mathcal D$ is defined by \end{aligned} \end{equation} -{\color{blue} By fixing the trace of the Hessian, we have effectively fixed -the value of the stability $\mu$ in all replicas to the value $\mu^*$.\\ +the value of the stability $\mu$ in all replicas to the value $\mu^*$. +\begin{itemize} + \item For $\mu^*<\mu_m$, this amounts to fixing the index density. Since the + overwhelming majority of saddles have a semicircle distribution, the + fluctuations are rarer than exponential. + \item For the gapped case $\mu^*>\mu_m$, there is a an exponentially small + probability that $r=1,2,...$ eigenvalues detach from the semicircle in such + a way that the index is in fact $N {\cal{I}}=r$. We shall not discuss + these subextensive index fluctuations in this paper, the interested reader + may find what is needed in \cite{Auffinger_2013_Complexity}. +\end{itemize} -$\bullet$ For $\mu^*<\mu_m$, this amounts to fixing the index density. The argument is that -the overwhelming majority of saddles have a semicircle distribution, the fluctuations are rarer than exponential. - -$\bullet$ For the gapped case $\mu^*>\mu_m$, there is a an exponentially small probability that $r=1,2,...$ eigenvalues detach -from the semicircle in such a way that the index is in fact $N {\cal{I}}=r$. -We shall not discuss these index fluctuations in this paper, the interested -reader may find what is needed in \cite{Auffinger_2013_Complexity} -} \subsubsection{The gradient factors} The $\delta$-functions in the remaining factor are treated by writing them in |